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Marja van den Heuvel-Panhuizen
Freudenthal Institute, Utrecht University, the Netherlands
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The nineties can be seen as the decade of standards. In many countries, at the governmental
level, decisions are made about what schools should teach their students. Examples of these
standards are the NCTM Standards in the United States, and the National Curriculum and the
Numeracy Project in the United Kingdom. This paper deals with the Dutch answer to this trend. It
will first give an overview of the main factors that determine the content of the primary-school
mathematics curriculum in the Netherlands. Subsequently, the focus will be on what is called a
“learning-teaching trajectory”; a new educational tool that intends to guide the teaching process.
After offering a short introduction to what is meant by a learning-teaching trajectory, an impression
will be given of the trajectory for calculations with whole numbers. This trajectory is the first one
that, commissioned by the Dutch Ministry of Education, has been developed by the TAL Project.
The project is carried out by the Freudenthal Institute of Utrecht University and the SLO2 and
conducted in collaboration with the CED3.
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Unlike many other countries, at primary school level the Netherlands does not have
centralized decision making regarding curriculum syllabi, text books or examinations (see Mullis et
al., 1997). None of these need approval by the Dutch government. For instance, the schools can
decide which textbook series they use. They can even develop their own curriculum. In general,
what is taught in primary schools is, for the greater part, the responsibility of teachers and school
teams and the teachers are fairly free in their teaching. To give some more examples, teachers have
a key to the school building, they are allowed to make changes in their timetable without asking the
school director (who often teaches a class too), and, as a last example, the teacher’s advice at the
end of primary school, rather than a test, is the most important criterion for allocating a student to a
particular level of secondary education.
Despite this freedom in educational decision making — or probably one should say thanks to
the absence of centralized educational decision making — the mathematical topics taught in
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primary schools do not differ much between schools. In general, all the schools follow roughly the
same curriculum. This leads to the question: what determines this curriculum?
Until recently, there were three important determinants for macro-didactic tracking in Dutch
mathematics education in primary school:
ƒ the mathematics textbooks series
ƒ the “Proeve”, a document recommending the mathematical content to be taught in primary
school, and
ƒ the core goals to be reached by the end of primary school as described by the government.
Since 1998, however, a fourth determinant is added. Before going into detail about this new
educational phenomenon, a short overview of the three factors that so far determined the content of
our mathematical curriculum in primary school will be provided .
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More than any other factor, in the Netherlands the textbook series influence what is taught to
the students. In the current world-wide reform of mathematics education, speaking about textbooks
— not to mention the use of them — often elicits a negative association. In fact, many reform
movements are aimed at getting rid of textbooks. In the Netherlands, however, the opposite is the
case. Here, the improvement of mathematics education depends largely on new textbooks. They
play a determining role in mathematics education. As a matter of fact one could say that textbooks
are the most important tools for guiding the teachers’ teaching. This is true for both content and
teaching methods, although with regard to the latter, the guidance provided is not sufficient to
reach all teachers. Several studies revealed indications that the intended reform that is aimed at by
Realistic mathematics Education4 is not optimally implemented in classroom practice (see
Gravemeijer et al., 1993; Van den Heuvel-Panhuizen and Vermeer, 1999).
The determining role of textbooks, however, does not mean that Dutch teachers are prisoners
of their textbooks. As already stated, Dutch teachers are fairly free in their teaching and the schools
can decide which textbook series they use.
Currently, over eighty percent of Dutch primary schools use a mathematics textbook series
which, to a greater or lesser degree, is inspired by RME. Compared to ten or fifteen years ago this
percentage has changed significantly. At that time, only half the schools worked with such a
textbook series (De Jong, 1986).
Presently, the most important textbook series are:
ƒ Pluspunt
ƒ De wereld in getallen
ƒ Wis en Reken
ƒ Rekenrijk
ƒ Alles Telt
ƒ Talrijk.
These textbook series are all developed by commercial publishers. The textbook authors are
independent developers of mathematics education, but they can use the ideas for teaching activities
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resulting from developmental research at, for instance, the Freudenthal Institute (and its
predecessors) and the SLO.
The textbook series are used by the teachers for planning their daily lessons. Activities are
mostly a mixture of whole class activities, individual work and group work. In general, the
differences between the content of the textbooks are not very large, but nevertheless there are also
some obvious dissimilarities.
The following pictures give an idea of how the textbooks look. Figure 1 contains a page from
the grade 2 “Pluspunt” booklet. The students work here in the domain of numbers up to one
hundred and it is clear that in this textbook series the number line is used as a model to support
calculations. The activity at the bottom of the page is about a number game. Here, the students have
to apply their knowledge in a “real” situation.
Figure 1 Page from the textbook series “Pluspunt”, grade2
Figure 2 Page from the textbook series “Wis en Reken”, grade 6
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Figure 2 contains a page from the sixth-grade “Wis en Reken” booklet. Here again, one can
see that attention has been paid to both problems with bare numbers and context problems. On the
lefthand page the students have to deal with information about the number of people who live in
each of the continents.
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An important aid in the development of textbooks is also the guidance which, since the mideighties, has come from a series of publications, called the “Proeve.”5 Treffers is the main author of
this series. The publications contain descriptions of the various domains within mathematics as a
primary school subject. Work on the “Proeve” is still going on and eventually there will be
descriptions for all the basic number skills, written algorithms, ratios and percentages, fractions and
decimal numbers, measurement, and geometry. Although the “Proeve” is written in an easy style
with many examples, it is not written as a series for teachers. Instead, it is intended as support for
textbook authors, teacher educators and school advisors. On the other hand, many such experts on
mathematics education were, and still are, significant contributors to the realization of this series.
Looking back at the Dutch reform movement in mathematics education, it can be concluded
that the reform proceeded in an interactive and informal way, without government interference.
Instead developers and researchers, in collaboration with teacher educators, school advisors and
teachers, worked out teaching activities and learning strands. Later on, these were included in
textbook series.
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Until recently there was no real interference from the Dutch government regarding the
content of educational programs. There was only a general law containing a list of subjects to be
taught. What topics had to be taught within these subjects was almost entirely the responsibility of
the teachers and the school teams. A few years ago, however, government policy changed and, in
1993, the Dutch Ministry of Education came up with a list of attainment targets, called “Core
Goals”. In 1998 a revised version was published (see OCandW, 1998) for mathematics containing
the following list of 23 goals, split into six domains (see Table 1). The goals describe what students
have to learn by the end of their primary school career (at age twelve). The content of the list is in
agreement with the “Proeve” documents previously mentioned.
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Table 1: Core goals for Dutch primary school students in mathematics
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Compared to goal descriptions and programs from other countries it is notable that some
widespread mathematical topics are not mentioned in this list, for instance, problem solving,
probability, combinatorics, and logic. Another striking feature of the list is that it is so simple. This
means that the teachers have a lot of freedom in interpreting the goals. At the same time, however,
such a list does not give much support to teachers. As a result, the list is actually a ‘dead’
document, mostly put away in a drawer when it arrives at school. Nevertheless, this first list of core
goals was important for Dutch mathematics education. The government’s publication of the list
confirmed and, in a way, validated the recent changes in the Dutch curriculum. The predominant
changes were:
ƒ more attention was to be paid to mental arithmetic and estimation
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ƒ formal operations with fractions were no longer in the core curriculum, the students now only
have to do operations with fractions in context situations
ƒ geometry was officially included in the curriculum
ƒ and also the insightful use of a calculator.
However, not all these changes have been included in the textbooks and implemented in
present classroom practice. This is especially true for geometry and the use of a calculator.
In the years since 1993, there have been discussions about these 23 core goals (see De Wit,
1997). Almost everybody is agreed that these goals can never be sufficient to support
improvements in classroom practice nor to control the outcome of education. The latter is
conceived by the government as a powerful tool for safeguarding the quality of education. For both
purposes, the core goals were judged as failing. Simply stating goals is not enough toachieve them.
For testing the outcome of education the core goals are also inappropriate. Many complaints were
heard that the goals were not formulated precisely enough to provide yardsticks for testing. These
arguments were heard not only for mathematics, but also for all the primary school subjects for
which core goals had been formulated.
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For several years it was unclear which direction would be chosen for improving the core
goals: either providing a more detailed list of goals for each grade expressed in operationalized
terms, or a description which supports teaching rather than pure testing. In 1997, the government
tentatively opted for the latter and asked the Freudenthal Institute to work out the description for
mathematics. This decision resulted in the start of the TAL Project6 in September 1997. The project
is carried out by the Freudenthal Institute and the SLO together, in collaboration with CED. The
learning-teaching trajectories that will be developed in this project are intended as the new guiding
factor for macro-didactic tracking in Dutch primary school mathematics education.
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The aim of the TAL Project is to develop learning-teaching trajectories for all domains of the
primary-school mathematics curriculum and to offer means for the implementation of these
learning-teaching trajectories in school practice. The preparation of the implementation includes
two activities. To begin with, for each learning-teaching trajectory an information module will be
devised for school teams. Secondly, modules about the learning-teaching trajectories will be
designed for an in-service course for teachers who are (or will become) mathematics coordinator at
their school.
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In total three learning-teaching trajectories will be developed: a trajectory for whole number
calculation, one for measurement and geometry, and one for fractions, decimals and percentages.
The project started with the development of a learning-teaching trajectory for whole-number
calculation. This first trajectory description for the lower grades (including K1, K2, and grades 1
and 2)7 was published in November 1998. The definitive version followed a year later (Treffers,
Van den Heuvel-Panhuizen, and Buys (Eds.), 1999). The following year the whole-number
trajectory for the higher grades of primary school (including grades 3 through 6) was published
(Van den Heuvel-Panhuizen, Buys and Treffers (Eds.), 2000). In 2001, both learning-teaching
trajectories were translated in English and published together in one book (Van den HeuvelPanhuizen (Ed.), 2001).
In 1999, a start was made with the development of a learning-teaching trajectory for
measurement and geometry. This will be finished in about one year.
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What is meant by a learning-teaching trajectory? To put it briefly, a learning-teaching
trajectory describes the learning process the students follow. It should not be concluded from this,
however, that it only contains the learning perspective. In our view, the term learning-teaching
trajectory8 has three interwoven meanings:
ƒ a learning trajectory that gives a general overview of the learning process of the students
ƒ a teaching trajectory, consisting of didactical indications that describe how the teaching can
most effectively link up with and stimulate the learning process
ƒ a subject matter outline, indicating which of the core elements of the mathematics curriculum
should be taught.
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A learning-teaching trajectory puts the learning process in line, but at the same time it should
not be seen as a strictly linear, singular step-by-step regime in which each step is necessarily and
inexorably followed by the next. A learning-teaching trajectory should be seen as being broader
than a single track. It is very important that such a trajectory description is doing justice to:
ƒ the learning processes of individual students
ƒ discontinuities in the learning processes; students sometimes progress by leaps and bounds
and at other times can appear to relapse
ƒ the fact that multiple skills can be learned simultaneously and that different concepts can be
in development at the same time, both within and outside the subject
ƒ differences that can appear in the learning process at school, as a result of differences in
learning situations outside school
ƒ the different levels at which children master certain skills.
In short, there is sufficient reason to talk about a learning-teaching trajectory having a certain
bandwidth.
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Compared to the goal descriptions that were traditionally supposed to guide education and
support educational decision making, the learning-teaching trajectory as it is worked out in the
TAL Project has some new elements that makes it a new educational phenomenon.
First of all, the trajectory is more than an assembled collection of the attainment targets of all
the different grades. Instead of a checklist of isolated abilities, the trajectory makes clear how the
abilities are built up in connection with each other. It shows what is coming earlier and what is
coming later. In other words, the most important characteristic of the learning-teaching trajectory is
its longitudinal perspectieve.9
A second characteristic is its double perspective of attainment targets and teaching
framework. The learning-teaching trajectory does not only describe the landmarks in student
learning that can be recognized en route, but it also portrays the key activities in teaching that lead
to these landmarks.
The third feature is its inherent coherence, based on the distinction of levels. The description
makes clear that what is learned in one stage, is understood and performed on a higher level in a
following stage. A recurring pattern of interlocking transitions to a higher level forms the
connecting element in the trajectory. It is this level characteristic of learning processes, which is
also a constitutive element of the Dutch approach to mathematics education, that brings
longitudinal coherence into the learning-teaching trajectory. Another crucial implication of this
level characteristic is that students can understand something on different levels. In other words,
they can work on the same problems without being on the same level of understanding. The
distinction of levels in understanding, which can have different appearances for different subdomains within the whole number strand, is very fruitful for working on the progress of children’s
understanding. It offers footholds for stimulating this progress.
The fourth attribute of the TAL learning-teaching trajectory is the new description format that
is chosen for it. The description is not a simple list of skills and insights to be achieved, nor a strict
formulation of behavioral parameters that can be tested directly. Instead, a sketchy and narrative
description, completed with many examples, is given of the continued development that takes place
in the teaching-learning process.
Giving the teachers a pointed overview of how children’s mathematical understanding can
develop from K1 and 2 through grade 6 and of how education can contribute to this development is
the main purpose of this alternative to the traditional focus on clear goals as the most powerful
engine for enhancing classroom practice. In no way, however, the trajectory is meant as recipe
book. It is, rather, intended to provide teachers with a “mental educational map” which can help
them, if necessary, to make adjustments to the textbook.
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To conclude this section a few words about how the TAL learning-teaching trajectories are
developed. The linking thread in the process is formed by the weekly discussions in the project
team, based on which the trajectory grew gradually. Input comes from a variety of sources:
analyses of textbook series, analyses of research literature, investigations in classrooms, and
consultations of experts.
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In the TAL trajectory for calculation with whole numbers, calculation is interpreted in a
broad sense, including number knowledge, number sense, mental arithmetic, estimation and
algorithms. In fact the trajectory description is meant to give an overview of how all these number
elements are related to each other. The next part of the paper will zoom in on this trajectory of
whole number calculation. To begin with, some general characteristics of the trajectory are
discussed. Consequently, some components of the trajectory are described more in detail.
From the trajectory for the lower grades some snapshots will be presented from addition with
numbers up to twenty and one hundred, and from the counting that precedes these operations. Here,
the spotlight will be on the coherence between the different levels of counting and calculating.
From the trajectory for the upper grades, as an example, attention will be paid to the topics of
estimation and column arithmetic and algorithms.
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As can be seen in Figure 3, the TAL trajectory for whole number calculation contains two
parts: one for the lower grades and one for the upper grades.
Figure 3 The TAL learning-teaching trajectory for whole number calculation in primary school
Although the learning-teaching process in both parts forms a continous process, it cannot be
neglected that each has its own characteristics. The students gradually come from a nondifferentiated way of counting-and-calculating to calculations in more specialized formats that are
dedicated to particular kinds of problems in a particular number domain. In other words, in the
lower grades, all activities with numbers can be generally labeled by “arithmetic”, whereas in the
upper grades different forms of calculations can be distinguished, like mental arithmetic,
estimation, column arithmetic, algorithms, and calculation by using a calculator.
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Another characteristic of the trajectory is the central role of mental arithmetic. It is seen as an
elaboration of the arithmetic work that is rooted in the lower grades and forms the backbone in the
upper grades.
Another feature of this trajectory is the explicit attention that is paid to numbers and number
relations. The idea is that if the students are familiar with the context of numbers, their position in
terms of magnitude and their internal structure, an important foundation is laid for the development
of their calculation abilities. The more the students know about numbers, the easier the problems
become for them. Or put it differently, if one invests in the numbers one gets the operations, so to
say, for free.
New in this trajectory is also that it contains a didactic for learning estimation. Although
estimation is considered as an important goal of mathematics education, in most of the textbooks a
structure for how to learn to estimate is lacking. The textbooks at most only contain some problems
on estimation, but doing some estimation problems from time to time is not enough to develop real
understanding in how an estimation works and it is not sufficient to comprehend what is possible
and what not when estimating.
Another novelty is the distinction that is made between algorithms and a less curtailed way of
calculating in which whole numbers are processed instead of digits, which is called “column
calculation”.
Finally, learning to calculate with whole numbers, of course, should include being able to use
a calculator. The trajectory describes how this ability can be built up, but at the same time it reflects
to be very cautious. The main goal is that the students can in the end make sensible decisions about
whether to use the calculator or not. Therefore, in the scheme, calculator use is placed between
parentheses.
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In the trajectory chapter about grade 1 the Restaurant lesson accomplishes a key role. This
lesson is set up to offer the students a learning environment in which they can developstrategies for
solving addition problems up to twenty in which they have to bridge the ten. The lesson that is
described in the TAL book was given in a mixed class containing K2 and grade 1 children, aged
five and six years. The teacher, Ans Veltman, is one of the staff members of the TAL team. She
also designed the lesson, although she would disagree with this — Ans feels that her student
Maureen was the developer of this lesson. Maureen opened a restaurant in a corner of the
classroom and everybody was invited to have a meal. The menu card shows the children what they
can order and what it costs. The prices are in whole guilders (see Figure 4).
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Figure 4: The menu card for Maureen’s restaurant
The teacher’s purpose with this lesson is. as said before, to work on a difficult addition
problem bridging the number ten. The way she does this, however, reflects a world of freedom for
the students. The teacher announced that two items could be chosen from the menu and asked the
children what items they would choose and how much this would cost. In other words, it appeared
as if there was no guidance from the teacher, but the contrary was true. By choosing a pancake and
an ice cream, costing 7 guilders and 6 guilders, respectively, she knew in advance what problem
the class would be working on; namely the problem of adding above ten, which is what she wanted
them to work on.
And there is more, there is a purse with some money to pay for what is ordered. The teacher
had arranged that the purse should contain five-guilder coins and one-guilder coins. This again
shows subtle guidance from the teacher. Then the students start ordering. Niels chooses a pancake
and an ice cream. Jules writes it down on a small blackboard. The other children shout: “Yeah ...
me too.” They agree with Niels’ choice. Then the teacher asks what this choice would cost in total.
Figure 5 shows a summary of what the children did.
Figure 5: The students’ strategies to solve 7+6
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Maureen counted 13 one-guilder coins. Six coins for the ice cream and seven coins for the
pancake. Thijs and Nick changed five one-guilder coins for one five-guilder coin and pay the ice
cream with “5” and “1” and the pancake with “5” and “1” and “1”. Then they saw that the two
fives make ten and the three ones make 13 in total. Later Nick placed the coins in a row: “5”, “5”,
“1”, “1”, “1”. Luuk came up with the following strategy: “First put three guilders out of the six to
the seven guilders, that makes ten guilders, and three makes thirteen.” Hannah did not make use of
the coins. She calculated: “6 and 6 makes 12, and 1 makes 13 guilders.” Another student came
with: “7 and 7 makes 14, minus 1 makes 13.”
This Restaurant lesson makes it clear that children who differ in skill and level of
understanding can work in class on the same problem. To do this, it is necessary that problems that
can be solved on different levels are presented to the children. The advantage for the students is
that sharing and discussing their strategies with each other can function as a lever to raise their
understanding. The advantage for teachers is that such problems can provide them with a crosssection of their class’s understanding at any particular moment. Such a cross-section includes the
different levels on which the students can solve the problem:
ƒ calculating by counting (calculating 7+6 by laying down seven on-guilder coins and six oneguilder coins and counting the total one by one)
ƒ calculating by structuring (calculating 7+6 by laying down two five-guilder coins and three
one-guilder coins)
ƒ formal (and flexible) calculating (calculating 7+6 without using coins and by making use of
one’s knowledge about 6+6).
The power of this cross-section is that it also offers the teachers a longitudinal overview of
the trajectory the students need to go along (see Figure 6). The cross-section of strategies at any
moment indicates what is coming within reach in the immediate future. As such, this cross-section
of strategies contains handles for the teacher for further instruction.
Figure 6 The cross-section shows “the future”
The next question – and this is really what the learning-teaching trajectory is about, namely
showing the coherence of the whole process – is what this calculating in grade 1 has to do with
resultative counting in Kindergarten and calculation up to one hundred in grade 2?
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Regarding the ability of resultative counting10 that the children attain in Kindergarten, again
different levels of working can be differentiated (see Figure 7). In the very beginning of the
children’s learning process – when the concept of number is not very thoroughly established – they
can meet difficulties in answering direct “how many” questions. To overcome this problem, a
context-related question can be asked instead, like:
ƒ how old is she (while referring to the candles on a birthday cake)?
ƒ how far may you jump (while referring to the dots on a dice)?
ƒ how high is the tower (while referring to the blocks of which the tower is built)?
In the context-related questions, the context gives meaning to the concept of number. This
context-related counting precedes the level of object-related counting in which children can handle
the direct ‘how many’ question in relation to a collection of concrete objects without any reference
to a meaningful context. Later on, the presence of the concrete objects is also no longer needed to
answer ‘how many’ questions. Via symbolizing, the children have reached a level of understanding
in which they are capable of what might be called formal counting, which means that they can
reflect upon number relations and that they can make use of this knowledge.
Figure 7 Different levels of counting (and calculating) in Kindergarten
Regarding calculation up to one hundred in grade 2 the following main strategies have been
distinguished in the learning-teaching trajectory:
ƒ the stringing strategy in which the first number is kept as a whole and the final answer is
reached by making successive jumps
ƒ the splitting strategy in which use is made of the decimal stucture and the numbers are split in
tens and ones and processed separately
ƒ the varying strategy in which use is made of knowledge of number relations and properties of
operations.
Examples of these different strategies for solving 48+29 are shown in Figure 8.
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The scheme in Figure 9 shows how the different calculation strategies are connected to each
other and that they all are based upon counting.
Figure 9 Coherence of the different levels of counting and calculating
And, of course, the feasibility of these strategies does not stop at the end of the lower grades.
The strategies keep their importance when the students, for instance, have to solve problems with
larger numbers and operations other than addition and subtraction. In Figure 10 the same basic
strategies can be recognized in the student work of grade 4 students when they had to explain how
they solved the problem about the coffee boxes.
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Figure 10 Strategies that grade 4 students applied for finding the number of coffee packets
The above examples of strategies show a clear longitudinal coherence of counting and
calculating. It explains how learning to calculate in one grade is connected to the learning process
in another grade. Insight into these levels in strategies provides teachers with a powerful mainstay
for gaining access to children’s understanding and for working on shifts in their understanding.
After starting, for instance, with context-related questions (“how old is she?”) the teacher can
gradually push back the context and reach the object-related questions (“how many candles are on
the birthday cake?”).
The level categories for counting and calculation problems and strategies differ remarkably
from, for instance, levels based on problem types11 and levels based on the size of the numbers to
be processed. They also deviate from the more general concrete/abstract distinctions in levels of
understanding12 and from level distinctions ranging from material-based operating with numbers to
mental procedures; with verbalizing as an intermediate state13. The ideas with which the TAL
levels in the early grades identify most can be found in the work of Donaldson (1978) and Hughes
(1986).14
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The usefulness and value of estimation is now widely acknowledged, but the didactics of
estimation is still a domain in development. The learning-teaching trajectory offers a first proposal
for a phased structure the students can go through for developing estimation skills. Therefore a
subdivision is made into the four sub-domains:
ƒ rounding off numbers
ƒ estimations in addition and subtraction
ƒ estimations in multiplication and division
ƒ estimations in case of incomplete data.
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The above sequence reflects, in a general sense, the trajectory followed by the children. The
basis of estimation is formed by rounding off numbers, which is followed by estimating in
calculation problems.
Of the four basis operations, addition and subtraction are offered first. Among other reasons,
this is because the consequences of rounding off for division and multiplication are often more
difficult to perceive. As long as the students are being asked to make only a very global estimate,
estimation in multiplication and division is indeed comparable with estimation in addition and
subtraction. This changes when a more refined estimation is involved and the students are also
required to indicate the magnitude of the deviation. This difference has mainly to do with the fact
that the effects of rounding off are not as clear cut in multiplication and division, because
deviations and imprecision become magnified. One effect of this magnification is that it becomes
more difficult to understand which type of rounding off results in the best estimates. This is
especially true when multiple numbers must be rounded off in a single problem. At this point, the
students have crossed into the terrain of the more skillful practitioner of arithmetic. Learning to
estimate in multiplication and division is a process that goes beyond primary school. The foregoing
makes that, generally speaking, addition and multiplication take a more important place in the
primary school curriculum for estimation than subtraction and division do.
Within the area of estimation two different types of estimation problems can be distinguished:
ƒ calculations with rounded off numbers
ƒ calculations with estimated values.
An example of the first type is the Bread problem (see Figure 11).
Figure 11 Bread problem
In this type of problems, in which only a global calculation is needed, the precisely given
numbers can be rounded off followed by an exact calculation with these round numbers.
The Arlette problem (see Figure 12) and the Apartment building problem (see Figure 13) are
examples of the second type of estimation problem.
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In this type of problems the necessary data is incomplete or unavailable. To solve such
problems the students have to good insight into the number system, and they should be familiar
with measures and sizes from daily life. Moreover the problems with estimated values often
requires making specific assumptions as a starting point.
The central theme of the learning-teaching trajectory in estimation is characterized by three
types of key questions that can be asked in order to elicit estimation and make this process of
estimation sensible. In fact, the following types of questions are suitable for this:
ƒ Are there enough?
ƒ Could this be correct?
ƒ Approximately how much is it?
It is these questions — which in themselves can take on all kinds of different forms — that
are the driving force behind learning to estimate and which, moreover, are anchored in estimation
as it occurs in everyday life. Although these three questions can be asked in every grade, the first
two types belong more to the initial phases of the learning process, while questions of the third type
should be offered later on. As a matter of fact, the latter type of question is a direct estimation
question where the children themselves must arrive at an estimation. The others are more indirect
estimation questions.
The most basic structure in the learning-teaching trajectory that guides the learning process is
the distinction in different phases in learning estimation:
ƒ the informal phase in which the students can globally determine answers without using the
standard rounding off rule
ƒ the rule-directed phase in which the students arrive at the standard rounding off rule for
operating with numbers and learn to apply this rule
ƒ the flexible and critical phase in which the students are capable of applying more balanced
estimation methods when operating with numbers and in which they can deal in a critical way
with rounded off and exact numbers.
In the informal phase the numbers are selected in such a way that a very global approach is
sufficient. It is sufficient if the children know that the total of 2,113 and 3,389 is more than 5,000.
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This becomes clear even if the students only look at the value of the largest position. Instead of
rounding off to the closest round number, the solution is then found by literally — or mentally —
covering the other position values with one’s thumb (see Figure 14).
Figure 14 Rounding off by only looking at the value of the largest position
By assigning problems where the very global method can lead to incorrect conclusions, the
students can arrive at the rule-directed phase. In such problems more precise rules about rounding
off are needed in order to come to a sensible answer, like, for instance, is the case in the Television
problem (see Figure 15).
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Figure 15 Television problem
By “cutting off “these numbers and only looking at the thousands, one could conclude that
the total is less than 10,000. However, if one slides one’s thumb to the right, then it becomes
immediately obvious that this estimation is too low; 800 plus 900 is more than 1000, which would
make the total more than 10,000.
The number of positions the students have to slide their thumbs to the right depends on the
specific problem. In this case, the positional values of the tens and ones do not matter. After all, as
one reaches the hundreds it becomes immediately clear that the total must be above 10,000.
During the phase of rule-directed rounding off, the children are expected to be able to round
off numbers when estimating in addition and subtraction problems according to the standard
rounding off rule.
The next step is the discovery that this rounding off rule does not always have to be strictly
applied when estimating in addition and subtraction problems. Then, the students arrive in the
phase of flexible and critical estimation. They realize that the standard procedure of rounding off
must be modified, especially if the numbers in an addition or subtraction problem are close to the
turning points of fifty, five-hundred, and so on. The Tickets problem (see Figure 16) makes this
obvious.
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If these numbers are rounded off according to the standard rule 3587 becomes 4 thousand,
2574 becomes 3 thousand, and 3928 becomes 4 thousand. The total is then 11 thousand, even
though 10 thousand is a better estimation. By comparing the children’s estimates and checking
them against the exact answer, refinements in the rounding off method can be brought up for
discussion.
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Another example of flexible and critical estimation is the Chicken problem (see Figure 16). It is
an striking example of what can go wrong when exact calculation is used with rounded off numbers.
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Figure 16 Chicken problem
At first glance it may seem strange that the reporter who wrote the newspaper clipping would
know exactly how many chickens had died, but it soon becomes clear how this total was arrived at.
Obviously, if only one chicken escaped and there were 26,000 chickens in the shed, this means that
25,999 were killed. Nonetheless, a serious mistake has been made in this calculation.
Although this is not explicitly stated in the newspaper article, the total of 26,000, of course,
stands for “approximately 26,000 chickens.” The number of chickens was probably rounded off to
the nearest thousand. This is why it is incorrect to subtract the single escaped chicken from this
total number. Children who understand how silly this calculation is will probably not have any
difficulty solving the Billion-million problem (see Figure 17).
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In order to understand that these answers are indeed correct, a certain amount of thinking is
necessary. The description “approximately 1 billion” indicates that this round number is in fact an
estimate. This is still reasonably simple to understand. It is more difficult to understand that this is
an estimation with a rounding off margin of one hundred million and that — for the total — it does
not matter whether one million is added or subtracted from the estimate of one thousand million. A
number line can help the children understand this.
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In the traditional teaching methods of primary school mathematics, the algorithms have a
high all-or-nothing character. Either the students will master them or they will not master them and
get stuck in the appointed procedures. It looks as if it is impossible to have in-between targets of
written calculations. The increasing magnitude of the numbers is often the only level distinction
that is made. Another failure of the mechanistic, rule-directed way of teaching algorithms is that
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part of the students has poor understanding of the procedures. For them the algorithms are a kind of
juggling with numbers. The main aim of the TAL trajectory for whole number calculation is to
make the process of learning algorithms more transparant and to offer different levels on which the
algorithms can be carried out. The foregoing is done by offering an alternative way of written
calculation which gives the students more grip on the procedure. It includes a less curtailed way of
written calculation in which whole numbers are processed instead of digits. This alternative way of
carrying out written calculations is called “column calculation”.
Figure 18 shows the attainment targets for written calculation for the end of primary school.
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the same for
Figure 18 Overview of the attainment targets for column calculation and algorithms
to be achieved at the end of primary school
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Characteristic for the column calculation procedure is that position values are used (four
hundred instead of four) and that the numbers are processed from left to right. In a way it is a form
of calculation that can be located between written and mental calculation. The interim results are
processed mentally (as is expressed in the “thought balloon”).
In particular for subtraction this column calculation procedure offers a new perspective in
written calculation. It breaks with the traditional procedure of “borrowing”. In the case of 845–382,
the students start with 800 minus 300 is 500. Then the tens are processed: 40 minus 80 means a
shortage of 40 (written down as –40). In the next step the ones are processed: 5 minus 2 is 3.
Finally, the interim results are used to get the final answer: 500 – 40 is 460; 460 + 3 is 463. This
way of carrying out subtractions looks at first glance rather unusual, but for the students it is ver y
natural way of doing. They often invent their own ways of notating (see Figure 21).
Figure 21 Student work of column subtraction with shortages
The overview of the attainment targets shows that for addition, subtraction, and multiplication
two possibilities are offered to the students as the final target to be achieved: column calculation or
algorithm calculation. With respect to division only column calculation is considered as an attainment
target for end primary school. In other words, the traditional algorithm is no longer compulsary. It is not
necessary that the students can do long division in the most curtailed way.
The reserved attitude towards algorithms that is reflected by the attainment targets should not
be misunderstood. Indeed, in a certain way they express the idea that students should not be
annoyed with senseless training of procedures which are not transparent for them and which do not
have practical use in real life. On the other hand, however, algorithms are valuable mathematical
tools of which the students should develop understanding. Therefore, the trajectory for column
calculations and algorithms does not stick to plainly carrying out procedures. Attention is also paid
to deepening the understanding and having fun with algorithms. For this the following type of
problems is used (see Figure 22).
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Figure 22 Special algorithm problems for in-depth understanding and fun
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So far some examples from the TAL learning-teaching trajectory for whole number
calculation. The purpose of this trajectory is that it can function as a guide for making didactical
decisions for mathematics teaching in primary school.
As already stated, the TAL Project is only just starting work on them. It is not yet known how
they will function in school practice and whether they can really help teachers. Inquiries made so
far (De Goeij, Nelissen, and Van den Heuvel-Panhuizen, 1998; Groot, 1999; Slavenburg and
Krooneman, 1999), however, give us a general feeling that they may indeed help teachers and that
the TAL learning-teaching trajectory on whole number arithmetic for the lower primary grades has
triggered something that in one way or another can bring not only the children but also Dutch
mathematics education to a higher level. For the TAL team it was interesting to discover that
developing a trajectory was not only a matter of writing down what was already known in a
popular and accessible way for teachers, but that the work on the trajectory also resulted in new
ideas emerging about teaching mathematics and revisiting our current thinking about teaching.
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