Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education

Abstract

This article focuses on a form of instructional design that is deemed fitting for reform mathematics education. Reform mathematics education requires instruction that helps students in developing their current ways of reasoning into more sophisticated ways of mathematical reasoning. This implies that there has to be ample room for teachers to adjust their instruction to the students' thinking. But, the point of departure is that if justice is to be done to the input of the students and their ideas built on, a well-founded plan is needed. Design research on an instructional sequence on addition and subtraction up to 100 is taken as an instance to elucidate how the theory for realistic mathematics education (RME) can be used to develop a local instruction theory that can function as such a plan. Instead of offering an instructional sequence that "works," the objective of design research is to offer teachers an empirically grounded theory on how a certain set of instructional activities can work. The example of addition and subtraction up to 100 is used to clarify how a local instruction theory informs teachers about learning goals, instructional activities, student thinking and learning, and the role of tools and imagery.

Full text

Learning Trajectories and Local Instruction Theories as Means of
Support for Teachers in Reform Mathematics Education
Koeno Gravemeijer,
Freudenthal Institute/
Department of Educational Research,
Utrecht University
Introduction
In the 1960’s and 1970’s theories for instructional design were en vogue in the educational-
research community. The most well known design theories from that period are probably
Gagné’s ‘Principles of Instructional Design’ (Gagné & Briggs, 1974). Since then, the interest for
instructional design seems to have faded away. More recently however, a renewed interest can be
noticed, especially in communities of mathematics educators. This relates to the current reform
efforts in mathematics education. Constructivism formed one of the catalysts of this reform
movement. Various interpretations of constructivism fueled the belief that mathematics education
should capitalize on the inventions by the students. This in part may explain why there was little
interest for instructional design within this reform movement, initially. We might even put it
stronger, by many, instructional design was seen as incompatible with mathematics education
that put the students’ own ideas and input at the forefront. This gradually changed, and the insight
is growing that mathematics education that aims to capitalize on the input of the students asks for
thorough planning. Although that would mean a different kind of planning than envisioned in
traditional instructional design strategies.
The principles of instructional design from the 1960’s and 1970’s do not fit reform
mathematics instruction. The main problem is that the older design principles take their point of
departure in the sophisticated knowledge and strategies of experts in order to construe learning
hierarchies. Following a so-called ‘task analysis’ approach the performance of the expert is taken
apart and laid out in small steps, and a learning hierarchy is constituted that describes what steps
are prerequisite and in what order these steps should be acquired. The result is a series of learning
objectives that may make sense from the perspective of the expert, but not from the perspective
of the learner. Let alone that there is any room for personal input from the learner.
What is needed for reform mathematics education, is a form of instructional design that
supports instruction that helps students to develop their current ways of reasoning into more
sophisticated ways of mathematical reasoning. For the instructional designer this implies a
change in perspective from decomposing ready-made expert knowledge as the starting point for
design to imagining students elaborating, refining, and adjusting their current ways of knowing.
This exactly is, and has been, the objective of many developmental-research projects at the
Freudenthal Institute in the Netherlands. These developmental-research projects focus on
developing so-called local instruction theories for various topics in mathematics (like addition
and subtraction, fractions, percents, area etc.).
Developmental Research
The core of developmental research, or design research as it is often called nowadays
(Edelson, 2002), is formed by classroom teaching experiments that center on the development of
instructional sequences and the instructional theories that underpin them (Gravemeijer, 1994,
1998). In the course of a classroom teaching experiment, the research team develops sequences
of instructional activities that embody conjectures about the course of students’ learning. To this
end, the designer conducts an anticipatory thought experiment by envisioning both how proposed
instructional activities might be realized in interaction in the classroom, and what students might
learn as they participate in them. Analyses of the actual process of students’ learning when the
Learning Trajectories and Local Instruction Theories 1

instructional activities are used in the classroom can then provide valuable information that can
be used to guide the revision of the instructional activities. The rationale for the instructional
sequence can be conceived as a local instructional theory that underpins a prototypical
instructional sequence (Gravemeijer, 1994, 1998). This (conjectured) local instruction theory can
be refined and revised in subsequent repetitions of the process of design and analysis, which can
become a cyclic process of its own.
The developmental research at the Freudenthal Institute grew out of the desire to develop
mathematics education that corresponds with Freudenthal’s (1973, 1991) ideal of “mathematics
as an human activity.” According to Freudenthal, students should be given the opportunity to
reinvent mathematics by mathematizing, mathematizing subject matter from reality and
mathematizing mathematical subject matter. In both cases, the subject matter that is to be
mathematized should be experientially real for the students. That is why this approach is named
“Realistic Mathematics Education” (RME). One of the core principles of RME is that,
mathematics can and should be learned on one’s own authority, through one’s own mental
activities.
Within the RME research community, one tries to answer the question of what mathematics
education--that would fulfill the above educational philosophy--should look like, by
experimenting with mathematics education in practice, and by reflecting on this experimental
practice. This reflection leads to the development of an educational theory, and this theory feeds
back into new experiments (fig. 1). This implies that the resulting theory, which we call: a
“domain-specific instruction theory for realistic mathematics education” (or RME theory, for
short), is always under construction.
PHILOSOPHY
EXPERIMENTAL
PRACTICE
RME THEORY
Figure 1. Cyclic process of theory development
In doing so, a research method emerged that was labeled “developmental research”
(Gravemeijer, 1994, 1998) in the Netherlands. Similar approaches emerged elsewhere, for
instance, under the names of “design experiments” (e.g. Brown, 1992), and “design research”
(e.g. Edelson, 2002). Mark that this research methodology emerges from a similar cyclic process
as the RME instruction theory. As a consequence, the research methodology has a similar
tentative character.1
Preliminary Design
Developmental research basically encompasses three phases: developing a preliminary
design, a teaching experiment, and a retrospective analysis. A classroom teaching experiment
typically begins with the clarification of mathematical learning goals and with an anticipatory
thought experiment in which the research team envisions how the teaching-learning process
might be realized in the classroom. This first step results in the explicit formulation of a
conjectured local instruction theory that is made up of three components: (a) learning goals for
Learning Trajectories and Local Instruction Theories 2

students, (b) planned instruction activities and the tools that will be used, and (c) a conjectured
learning process in which the research team anticipates how students’ thinking and understanding
might evolve when the instructional activities are used in the classroom. This planning process
acknowledges the importance of listening to students and assess their current understandings, but
also stresses the importance of anticipating the possible process of their learning as it might occur
when planned but revisable instructional activities are used in the classroom. The developmental
researcher may take ideas from whatever sources to construe an instructional sequence. The
sources may be curricula, texts on mathematics education, research reports and the like. Mark,
however, that adopting often means adapting an activity in such a manner that the instructional
activities are being detached from their original context. The way by which this is done is
determined by the researcher’s overall vision on mathematics education. This way of working
may be described as “theory-guided bricolage” (Gravemeijer, 1994), since it resembles the
manner of working of what the French call a “bricoleur.” A bricoleur is an experienced
tinker/handy man, who uses as much as possible those materials that happen to be available. To
do so, many materials will have to be adapted. Moreover, the bricoleur may have to invent new
applications that differ from what the materials were designed for. The developmental researcher
follows a similar approach, but the way in which selections and adaptations are made, will be
guided by a theory--in our case, RME theory.
RME theory
One of the central tenets of RME is that the starting points of instructional sequences should
be experientially real to students in the sense that they can immediately engage in personally
meaningful mathematical activity (Gravemeijer 2000; Streefland, 1990). As a point of
clarification, we should stress that the term “experientially real” means only that the starting
points should be experienced as real by the students, not that they should necessarily involve real
world situations. Thus, arithmetic tasks presented by using conventional notation might be
experientially real for students for whom the whole numbers are mathematical objects.
A second tenet of RME is that in addition to taking account of students’ current
mathematical understandings, the starting points should also be justifiable in terms of the
potential mathematical end points of the learning sequence. This implies that students’ initially
informal mathematical activity should constitute a basis from which they can abstract and
develop increasingly sophisticated mathematical concepts as they participate in classroom
mathematical activities. At the same time, the situations that serve as starting points should
continue to function as paradigm cases that involve rich imagery and thus anchor students’
increasingly abstract mathematical activity.
Although these central tenets give a sense of direction, they do not tell the designer how to
fulfill them. But several decades of developmental/design research at the Freudenthal Institute in
the Netherlands resulted in a domain-specific instruction theory that is grounded in numerous
concrete elaborations of the RME approach (Gravemeijer, 1994; Streefland, 1990; Treffers,
1987). When interpreting this domain-specific instruction theory as an instructional design
theory, we can point to three design heuristics: guided reinvention (Freudenthal, 1973), didactical
phenomenological analysis (Freudenthal, 1983), and emergent modeling (Gravemeijer, 1999).
These design heuristics help the designer/researcher in designing a conjectured learning
route together with a set of potentially useful instructional activities that fit this learning route.
More in detail, this implies that the designer/researcher will think through what mental activities
of the students may be expected when they engage in the instructional activities, and how those
mental activities may help the students developing the envisioned mathematical insights. In the
teaching experiment, those conjectures are put to the test.
The Teaching Experiment
To clarify this process of anticipating and testing--that is central to the teaching experiment--
Learning Trajectories and Local Instruction Theories 3

it may be helpful to first elaborate on Simon's (1995) analysis of the deliberations of a
"constructivist teacher", who is engaged in the task of selecting and presenting instructional
activities. Simon analyses his own role as a teacher in a small classroom teaching experiment in
terms of a process of decision making about content and task. His analysis shows a teacher who
is constantly thinking about what the students might be thinking, and how he could influence
their thinking in an indirect manner. He tries to envision the mental activities of the students as
they work the problems he might pose to them, and he tries to anticipate how their thinking might
help them to develop the insights he wants them to develop.
To describe this role of the teacher, Simon introduces the notion of a “hypothetical learning
trajectory” (HLT):
“The consideration of the learning goal, the learning activities, and the thinking and
learning in which the students might engage make up the hypothetical learning
trajectory (...).” (Simon, 1995, 133)
The key in this learning trajectory is in the thinking that the students might engage in as they
participate in the instructional activities, which the teacher has in mind. He speaks of a
hypothetical learning trajectory because the actual learning trajectory is not knowable in advance.
Nevertheless, although individual learning trajectories may vary, learning often proceeds along
similar paths. The teacher therefore can constitute a hypothetical learning trajectory based on
expectations about such paths. The actual constitution of the instructional activities in the
classroom and course of the teaching-learning process offer the teacher opportunities to find out
to what extend the actual learning trajectories of the students correspond with the hypothesized
ones. This will lead to new understandings of the students’ conceptions. These new insights and
the experience with the instructional activities as such, will form the basis for the constitution of a
modified hypothetical learning trajectory for the subsequent lessons. Simon (1995) describes this
repetition of designing hypothetical learning trajectories and enacting and analyzing the
corresponding instructional activities as a “mathematics teaching cycle”.
Experimenting in the Classroom
A similar cyclic process of thought experiments and instruction experiments (see also
Freudenthal, 1991) forms the backbone of the developmental research method employed in the
teaching experiment (see fig. 2).
thought
exp.
thought
exp.
thought
exp.
thought
exp.
thought
exp.
instruction
exp.
instruction
exp.
instruction
exp. instruction
exp.
Figure 2. Developmental research, a cumulative cyclic process.
The process that governs this research process is very similar to that of the mathematical
teaching cycles as described by Simon (1995). Even though the goal of the researcher is not to
solve an immediate problem, but to constitute a well-considered, and empirically grounded,
Learning Trajectories and Local Instruction Theories 4

instruction theory. Simon coined the term hypothetical learning trajectory in the context of a
teacher preparing the next-day’s instructional activity. But in a similar manner, a teacher--or an
instructional designer for that matter--may envision a more encompassing learning route. Of
course, when enacting the corresponding instruction, the teacher will have to adjust time over
time to what actually happens in the classroom. But the overall plan provides a basis for thinking
through how to get back on track. In relation to this, we may lend Simon’s (1995) journey
metaphor. When making a journey we may start out with a well-considered travel plan.
Nevertheless, our actual journey will differ from this plan, due to the circumstances that we meet
during our journey. Moreover, not every aspect of a journey can be planned in advance. A
preconceived conjectured learning route may be compared with a travel plan, where what’s
actually going on in the classroom constitutes the journey.
The result of a developmental-research experiment can be cast in terms of a grounded theory
on a learning route. The latter is also referred to as a conjectured local instruction theory
(Gravemeijer, 1994, 1998). The term local instruction theory is used to convey the intend of
offering more than a description of a learning route, or the corresponding instructional activities.
With a local instruction theory we also want to offer a rationale. In contrast with traditional
instructional-design research, the objective of developmental research is not to offer an
instructional sequence that “works”, but to offer the user a grounded theory on how the
researchers/designers think that a certain set of instructional activities might work. This appears
to be in line with the shift in norms of justification that is observed by the RAC of the NCTM
(1996). They add, that in case of a justification in terms of a theory about how a new approach
works, the teachers have a basis for thinking through what adaptation might be needed to make
the new approach work in their classroom.
Returning to the role of the conjectured local instruction theory, we could say that the local
instruction theory offers a framework against the background of which, the teachers can develop
hypothetical learning trajectories. For, how local the instruction theory may be it will still be
general and unspecified theory in relation to the concrete decisions that teachers have to make in
actual classrooms. The local instruction theory will not be tailored to the specific situation in
classroom X of teacher Y. This creates a need for the teacher to develop hypothetical learning
trajectories that fit their own situation, while using the local instruction theory as a resource.
In the developmental research project, the mathematical teaching cycles serve the
development of the local instruction theory. In fact there is a reflexive relation between the
thought and instruction experiments, and the local instruction theory that is being developed. At
one hand, the conjectured local instruction theory guides the thought and instruction experiments,
and at the other hand, the microinstruction experiments shape the local instruction theory (fig. 3).
C O N J E C T U R E D L O C A L I N S T R U C T I O N T H E O R Y
t h o u g h t
e x p .
t h o u g h t
e x p .
t h o u g h t
e x p .
t h o u g h t
e x p .
t h o u g h t
e x p .
i n s t r u c t i o n
e x p .
i n s t r u c t i o n
e x p .
i n s t r u c t i o n
e x p .
i n s t r u c t i o n
e x p .
Learning Trajectories and Local Instruction Theories 5

Figure 3. Reflexive relation between theory and experiments.
In order to be able to adjust the envisioned instructional activities on a daily basis, it is
desirable that the researchers be present in the classroom every day while the teaching
experiment is in progress. The ongoing analyses of individual children’s activity and of
classroom social processes inform new anticipatory thought experiments in the course of which
conjectures about possible learning trajectories are frequently revised. As a consequence, there is
often an almost daily modification of local learning goals and instructional activities.
This focus on the ongoing process of experimentation emphasizes that ideas and conjectures
are modified while interpreting students’ reasoning and learning in the classroom. The empirical
data on the activities of the students are interpreted in light of the theoretical framework of RME.
In addition to this, the interpretive framework of Cobb & Yackel (1996) can be used to help the
researchers make sense of classroom events.
Retrospective Analysis
The results of design experiments can not be linked to pre and post tests results in the same
direct manner as is common in standard formative evaluation, since the proposed local instruction
theory and prototypical instructional sequence will differ from what is tried out in the classroom.
Because of the cumulative interaction between the design of the instructional activities and the
assembled empirical data, the intertwinement between the two has to be unraveled to pull out the
optimal instructional sequence in the end. For it does not make sense to include activities that did
not match their expectations, but the fact that these activities were in the sequence will have
affected the students. Therefore adaptations will have to be made when the non-, or less-
functional activities are left out.
Consequently, the instructional sequence will be put together as a reconstruction of a set of
instructional activities, which are thought to constitute the effective elements of the sequence.
This reconstruction of the optimal sequence will be based on the deliberations and the
observations of the developmental researcher(s). In this manner, the result of a developmental-
research experiment will be well considered and empirically grounded2.
Core Elements of Learning Trajectories/ Local Instruction Theories
The point of departure in this article is, that if you really want to do justice to the input of the
students and want to build on their ideas, then you need a very well founded plan. At first sight
there may seem to be a contradiction between adapting your teaching to the ideas and suggestions
of the students, and planning your instruction in advance. This, of course, depends on what your
planning entails. Here we may take Simon’s (1995) study of the role of a “constructivist” teacher
as a paradigm. Together with his notion of mathematical teaching cycle, the hypothetical learning
trajectory offers us a handle on how to support students’ learning with an eye on possible
endpoints, while at the same time building on the students’ own thinking. At the one hand, the
teacher engages the students well-considered instructional activities on base of the anticipation of
certain mental activities that are perceived as desirable, while, at the other hand, knowledge of
these anticipated mental activities and their role in a conjectured learning process, enables the
teacher to evaluate the students’ activities and to adjust the learning trajectory to the outcome of
such evaluations. In this manner, the hypothetical learning trajectory presents itself as the pre-
eminent tool for enacting instruction that puts the students’ understanding and thinking to the
forefront. Key in this process is the teacher’s role in choosing the instructional activities in light
of the current thinking of the students. It is by this tuning that the teacher can anticipate the
students’ thinking.
Learning Trajectories and Local Instruction Theories 6

In the following, we want to discuss the core elements of a local instruction theory, on the
basis of an instructional sequence that is developed in a teaching experiment in Nashville (TN) by
Cobb, Gravemeijer, McClain and Stephan of Vanderbilt University. But, before zooming in to the
instructional sequence, we want to stress the importance of the classroom culture that is essential
for the enactment of an instructional sequence. To realize a problem-centered, or inquiry-based
learning process, certain classroom social norms (Cobb & Yackel, 1996) need to be established.
Such social-norms may include expectations and obligations regarding explaining and justifying
solutions, attempting to make sense of explanations given by others, indicating agreement and
disagreement, and questioning alternatives in situations where a conflict in interpretations or
solutions has become apparent.
In addition to this certain socio-math norms have to be established to create the opportunity
for the students to evaluate mathematical progress.
We further want to caution that the way the instructional sequence is described may evoke
an image of the learning process of one single student, or of a classroom learning in unison. In
practice, however, individual learning routes will vary, and, more importantly, the interaction
between students, both in group work and in whole-class discussion, is essential for both the
progress of the group and of individual students. In relation to this, we want to argue that there is
a central role for the teacher in orchestrating discussions, and in framing topics for discussion.
Mark that our goal is not that students will come to appreciate the superiority of the most
sophisticated solution. Instead, the objective is to organize discussions on issues that are judged
mathematically significant, and relevant for the students’ mathematical development (see Cobb,
1997).
Exemplary Instructional Sequence
The goal of the instructional sequence we want to use as an example is to foster the use of
flexible mental computation strategies for addition and subtraction up to one hundred.3 This does
not mean, however, that the idea is to teach the students a set of strategies. Instead, we want the
students to develop a framework of number relations that offers the building blocks for flexible
mental computation. To solve 35+29, for instance, one may use number relations such as 35 = 30
+ 5, 35 + 5 = 40, 35 + 10 = 45, 35 + 20 = 55, or 35 + 30 = 65 and 29 = 30 -1.
Mark that, as the example shows, decuples will play a central role in this framework of
number relations.
Cardinal and Ordinal Aspects
Research shows that the strategies students use to solve addition and subtraction problems
up to 100 fall in two broad categories (Beishuizen, 1993), which we may denote “splitting” and
“counting”.
A task like 44 + 37, for instance, may be solved in the following manner,
- by splitting tens and ones:
44 + 37 = …; 40 + 30 = 70; 4 + 7 = 11; 70 + 11 = 81, or
- by counting in jumps:
44 + 37 = …; 44 + 30 = 74; 74 + 7 = 81, or:
44 + 37 = …; 44 + 6 = 50; 50 + 10 = 60; 60 + 10 = 70; 70 + 10 = 80; 80 + 1 = 81,
or via some other combination of jumps of tens and ones.
Beishuizen (1993) found that procedures based on splitting tens and ones led to more errors,
than solution procedures that were based on counting on and counting back. The latter type
solution procedures were therefore taken as a point of departure for the instructional sequence
under discussion. This choice was connected to the observation that students tend come up with a
wide variety of counting solutions when confronted with “linear-type” context problems. In this
Learning Trajectories and Local Instruction Theories 7

respect, we may distinguish linear-type situation from collection-type situations that lend
themselves more to a splitting of tens and ones.
A closer look at counting strategies shows that these strategies rely on integrating the
cardinal aspect of number (quantity) and the ordinal aspect of number (position/rank). Most
addition and subtraction problems concern quantities, while the solution procedures consist of
moving up and down the number sequence. We would argue that it is important that the students
connect the first and the latter.
Empty Number Line
The objective to help students make a connection between the cardinal and the ordinal aspect
presents an argument to introduce the number line as a tool. From an expert’s point of view, a
number line integrates both the cardinal aspect (line segment) and the ordinal aspect (point). In
addition to this, the number line offers a way of symbolizing that fits nicely the various counting
strategies--by describing the subsequent counting steps as arcs on an empty number line
(Gravemeijer, 1994, 1999). We speak of an empty number line, since this number line is empty
except for the numbers that are actually needed, and added by the students a part of the solution
process (see fig. 4).
35 6460
4
40 50
510 10
Figure 4. 64-29 on the empty number line.
This interpretation of the number line, however, is not self-evident. For the students it does
not speak for itself what the marks on the number line signify. We may illustrate this with the
following classroom episode.
This classroom episode was part of an earlier teaching experiment, within which we tried to
build on activities known under the label “The Candy Shop” (see Cobb e. a., 1997). The
instructional intent of the candy shop sequence is to support the students’ initial coordination of
units of ten and one. The story revolves around the production and sale of candies that are
represented by unifix cubes. Within this scenario, the notion of packing candies into rolls of 10 is
developed as a convention. Next the students are asked to generate different partitionings of a
given collection of candies, and to solve problems that involve incrementing and decrementing
numbers of candies that ask for packing and unpacking candies.
At the end of the Candy Shop sequence the empty number line was introduced in a whole
class mental computation activity called Target. In this activity, the task of the students is to
figure out how much one would have to change a given number to reach a fixed target number.
During the Target Activity, the ways in which the students solved these problems were
symbolized by drawing number lines (see fig. 5).
Learning Trajectories and Local Instruction Theories 8

6
12
13
6
1
6 + 6 = 12
6 + 7 = 13
Figure 5. Symbolizing a solution for 6+…=13 in the context of the Target activity.
The empty number line was subsequently extended to 100 during an activity in which it
served as a record of simulated transactions in the candy shop. The hypothesis that the empty
number line could be grounded in the candy-shop activities, initially appeared to be confirmed by
the facility with which the children, working in pairs, created number lines to tell stories about
what happened in the candy shop. However, later events in the classroom led us to revise our
view. Problems arose when the teacher posed addition and subtraction tasks by drawing a
horizontal empty number line, and by describing transactions in the candy shop, but without
acting them out. Here it showed that the students interpreted the empty number line in a variety
of different yet personally meaningful ways.
82 90
88 89
1 1
8
Figure 6. Counting down.
The differences in interpretation can be illustrated with figure 6. As part of a counting back
strategy for solving 90 - 8, some of the students interpreted this figure as showing that you have
taken away three candies: 90, 89, and 88. Other students argued that you had taken away two
candies.
In retrospect, we concluded that the students adopted the empty number line primarily as a
way of notating, and not as a way of modeling a (mental) activity. We inferred that this
calculational interpretation is stimulated by the fact that the students conceptualized the Candy-
Shop activities from a collection perspective, and not from a linear/counting perspective. In the
Candy-Shop setting, the students were routinely structuring quantities in tens and ones. This
“split” in tens and ones then guided their solution methods, while the empty-number-line
representation does not lend itself for this type of solution methods.
We may contrast this with the proposals of Whitney (1988) and Treffers (Treffers & De
Moor, 1990), who suggest to start with various ways of counting beads on a bead string as a lead
in to use of the number line. The proposed bead string of 100 beads that consists of alternating
groups of ten light and ten dark beads (fig .7).
Learning Trajectories and Local Instruction Theories 9

Figure 7. Counting on a bead string, 38 and 24 more.
What this approach aims for is that the students model situation-specific solution methods
that are situated in the activity of manipulating with the bead string. In this situation, we may
expect the students to develop solution methods of a linear type; like counting on/counting down
in bigger or smaller jumps While we may expect them to start using the decuples as reference
points. This activity is being modeled with arcs on the empty number line. Consequently, later
on, the past experience of acting with the bead string provides the necessary imagery behind the
jumps on the number line.
In the aforementioned teaching experiment this imaging was lacking. As a consequence, the
children had to create a semantic grounding for the marks on the empty number line individually
and privately by interpreting it in terms of counting activity. However, they did this in a variety
of different ways, which made it impossible for them to communicate effectively about what has
previously been an unproblematic activity.
One of the underlying problems is that a hash mark with a number, like 90 for instance, can
be thought of as signifying 90 (candies, beads,) or the 90th (candy or bead).
The Ruler as a Model
We realized ourselves that a similar issue plays in measurement. Transposed to a
measurement situation, the question might be: What does the number 90 on a ruler signify? It
may be noted that this relates to a conceptual understanding of measurement. Or to put it
differently, from a mathematical point of view, it is not sufficient for the students to learn to
measure accurately. Students will have to come to interpret the activity of measuring as the
accumulation of distance (cf. Thompson & Thompson 1996). The latter implies that each
number word used in the activity of iterating signifies the total measure of the distance measured
until that moment. Another aspect of a conceptual understanding is that the students conceive the
results of measuring as structured quantities. E.g. a distance of 20 units can be thought of as
being composed of 20 measurement units, or as composed of two distances of ten units, or of
distances of five units and fifteen units, and so forth.
Speculating on the genesis of the ruler in history, we reasoned that we could take the view
that the ruler came about as a curtailment of iterating a measurement unit. So the ruler could be
thought of as a model of iterating some measurement unit. From earlier teaching experiments we
already knew that the empty number line could function as a model for mathematical reasoning in
the context of mental computation strategies for reasoning with numbers up to 100. The
connection between the two would be in the relation between iterating measurement units as
accumulating distances, and a cardinal interpretation of positions on the number line.
We designed an instructional sequence that may be crudely described by the following series
of steps:
(1) The students start with measuring various lengths by iterating some measurement unit.
(2) This measuring activity is curtailed to measuring with tens & ones.
(3) The activity of iterating tens & ones is modeled with a ruler.
Learning Trajectories and Local Instruction Theories 10

(4) Next, the activity of measuring is extended to incrementing, decrementing and comparing
lengths. Here the activity shifts from measuring to reasoning about measures. This type
of tasks will offer the students the opportunity to develop arithmetical solution methods
that may be supported by referring to the decimal structure of the ruler.
(5) These arithmetical solution methods can be symbolized with arcs on a schematized ruler.
(6) This schematized representation is used as a way of scaffolding, and as a way of
communicating solution methods for all sorts of addition and subtraction problems.
In practice, the measuring activities are more refined, and consist of the following activities:
- measuring by pacing, heel to toe,
- measuring by iterating a paper 'footstrip' of five feet,
- measuring with unifix cubes (denoted as 'smurf food cans'),
- measuring with a bar of ten unifix cubes (called 'smurf bar'),
- measuring with a paper strip of 10 unifix cubes,
- measuring with a measurement strip/ruler construed out of ten paper strips.
Reflection
To reflect on this instructional sequence we may start by recalling that a conjectured local
instruction theory that is made up of three components: Learning goals for students, planned
instruction activities and the tools that will be used, and a conjectured learning process in which
one anticipates how students’ thinking and understanding might evolve when the instructional
activities are used in the classroom. Key elements here, are: the learning goals, the activities, and
the thinking and learning of the students. We will take those elements as our point of departure
for a reflection, while we will give special attention to the role of tools.
Goals
The goal of helping the students to develop a framework of number relations sets the stage.
Especially since the number relations will have to be grounded in quantitative meaning, and are
not to be acquired as abstract rules. Therefore, the objective is to promote that students will start
to generalize over concrete instances of quantitative relations (e.g. 4 feet and 4 feet more equals 8
feet) to construe mathematical relations (like 4+4=8). The character of this goal already points to
a certain educational approach. Here a task analysis followed by teaching individual learning
steps would not be very appropriate. Instead, opportunities will have to be created for the students
to generalize and to construe the mathematical relations themselves. However, we would argue
that the exemplary sequence shows that it is quite possible to map out a route along which the
students can construct the intended knowledge (via a process of guided reinvention). We will
substantiate this claim by describing how the learning process of the students evolves under
influence of their participation in the instructional activities.
Activities, and Thinking and Learning
The instructional sequence starts with measuring lengths by pacing. Here, measuring
emerges as quantifying space. That is to say, the result of pacing signifies a sequence of paces
that can be counted for the students. While, the last number word initially signifies last pace
rather than accumulation of distances covered by all paces. In this phase, measuring is still
inseparable from the bodily act of pacing. Later, measuring becomes divorced from activity. The
students reorganize their understanding, and the extension of an item starts to take precedence the
activity. Measuring with collections of fives (footstrip) and tens (smurf bar) respectively, ask for
the coordination of measuring with two measurement units. It is in the process of coming to grips
with this coordination, that measuring starts to signify the accumulation of distances covered by
all iterations. Eventually, the result is partitioned space structured in 10s and 1s. From then on it
is as if students are acting in an environment in which items are already partitioned, they already
Learning Trajectories and Local Instruction Theories 11

have a measure. So the measurement strip can be used to simply specify the measure. Next, when
the measure of an item signifies an objective property of an item for the students, the students can
take a measure as a given to specify an item’s spatial extent. Then tasks about incrementing,
decrementing, and comparing lengths can be solved with help of the measurement strip. E.g. the
students may count spaces on the measurement strip to specify the measure of the spatial
extension between two lengths. Gradually, however, arithmetical reasoning takes the place of
measuring, which eventually can be described with arcs on the empty number line.
When making this transition, it is essential that the students differentiate between the activity
of measuring, and the activity of representing arithmetical strategies. On the empty number line
you want the students to express how they would, for instance, increment 64 with 28. E.g., by
first measuring 64, then add 6 ones which would get one to 70, then measure two times ten which
would get one to 80, and 90 respectively, and finally add two ones which gets one to 92. When
describing this method, it would be sufficient to show that you start at 64, add 6, arrive at 70, then
add 10 arriving at 80, another 10 arriving at 90, and 2 arriving at 92. To try to strive for an exact
proportional representation of all the jumps would severely hamper a flexible use of the number
line. Therefore we will have to make sure that the students are aware of the distinction between
the ruler as a measurement tool, and the empty number line as a means of describing solution
procedures. Thus when they make drawings, the intention of the students should not be to make a
schematic drawing of a measuring device, but to make a drawing that shows their arithmetical
reasoning.
What is expected is, that in the course of the sequence, a shift is taking place in which the
student’s view of numbers transitions from referents of distances to numbers as mathematical
entities. This shift involves a transition from viewing numbers as tied to identifiable objects or
units (i. c. numbers as constituents of magnitudes; “37 feet”) to viewing numbers as entities on
their own (“37”). For the student, a number viewed as a mathematical entity still has quantitative
meaning, but this meaning is no longer dependent upon its connection with identifiable distances,
or with specified countable objects. In the student’s experienced world, numbers viewed as
mathematical entities derive their meaning from their place in a network of number relations.
Such a network may include relations such as 37=30+7, 37=3x10+7, 37=20+17, 37=40-3. The
critical aspect of this network is that the students’ understanding of these relations transcends
individual cases. That is, when students form notions of mathematical entities, they come to view
relations like the above as holding for any quantity of 37 objects (including a magnitude of 37
units).
In this manner, the goal developing a framework of number relations, which is grounded in
quantitative meaning, is reached.
Tools and Imagery
The general orientation of helping students to develop their current ways of reasoning into
more sophisticated ways of mathematical reasoning has implications for the role of tools as well.
Ideally, the students should invent the necessary tools for themselves. This, however, is not really
feasible. We try nevertheless to ensure that the tools in a sense emerge from the activity of the
students. We try to do so by requiring that the use of new tools be grounded in some imagery for
the students. That is to say, that there has to be some history in the learning process of the student
that renders meaning to the activity with a new tool. In the case of the arcs on the number line, it
is the activity of jumping along the ruler while using decuples as reference points that offers the
imagery, and gives meaning to the numberline activity. In a similar manner, the activity of
measuring by iterating units of ten and one (unifix cube(s)) offers a background for reading of
measures from a ruler. Similar relations can be found in a more detailed analysis of the sequence
of tool use and tool invention in the sequence (feet, footstrip, unifix cubes, smurf bar, paper strip,
measurement strip) (see Stephan, Bowers, Cobb, & Gravemeijer, in press). In relation to the
Learning Trajectories and Local Instruction Theories 12

latter, we already noted that the students do not literally invent those tools. We take care,
however, that the students are involved in the invention process. This can be done by a careful
introduction of each new tool according to the following set up: Each new tool has to come to the
fore as a solution to a problem (e.g. how to measure more efficiently), the students are given the
opportunity to think about a solution to that problem for themselves, on the basis of which the
students may decide to accept the new tool as an acceptable solution to the problem. In this
manner, the students will experience an involvement in the invention process even though they
do not invent the tools for themselves.
Conclusion
With this brief example we hope to have given an impression, what role hypothetical
learning trajectories and local instruction theories can play in offering teachers support in
realizing mathematics instruction that can do justice to the input of the students and help them
build on their own ideas. In connection with this we have shown what role instructional design
has to offer for reform mathematics education. In relation to this we presented developmental
research, or design research, as an alternative for the instructional design theories of the 1960’s
and 1970’s. Developmental research is not aimed at offering teachers scripted lessons, but tries to
offer teachers rationales (in the form of local instruction theories) and resources. This then
enables teachers to make informed decisions, while supporting students in developing their
current ways of reasoning into more sophisticated ways of mathematical reasoning
Literature
Beishuizen, M. (1993). Mental Strategies and materials or models for addition and
subtraction up to 100 in Dutch second grades. Journal for Research in Mathematics Education,
24 (4), 294-323.
Brown, A.L. (1992). Design experiments: Theoretical and methodological challenges in
creating complex interventions in classroom settings. Journal of the Learning Sciences, 2, 63-
112.
Cobb, P. (1997). Instructutional design and reform: a plea for developmental research in
context. In: Beishuizen, M., Gravemeijer, K.P.E., & Lieshout, E.C.D.M van (Eds.) The Role of
Contexts and Models in the Development of Mathematical Strategies and Procedures. Utrecht:
Cdß Press, 273-289.
Cobb, P. & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the
context of developmental research. Educational Psychologist, 31, 175-190.
Cobb, P., Gravemeijer, K., Yackel, E., McClain, K., and Whitenack, J. (1997).
Mathematizing and Symbolizing: The Emergence of Chains of Signification in One First-Grade
Classroom. In: D. Kirschner, and J.A. Whitson (Eds.). Situated cognition theory: Social,
semiotic, and neurological perspectives. Hillsdale, NJ: Erlbaum, 151-233.
Edelson, D. C. (2002). Design Research: What We Learn When We Engage in Design.
Journal of the Learning Sciences, 11, 105-121.
Gagné, R.M. and L.J. Briggs (1979). Principles of instruction design (2nd ed.). New York:
Holt, Rinehart and Winston Inc.
Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht, The
Netherlands: CD-β Press.
Gravemeijer, K. (1998). Developmental Research as a Research Method, In: J. Kilpatrick
and A. Sierpinska (Eds.) Mathematics Education as a Research Domain: A Search for Identity
(An ICMI Study). Dordrecht: Kluwer Academic Publishers, book 2, 277-295.
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal
mathematics. Mathematical Thinking and Learning.1 (2), 155-177.
Freudenthal, H. (1973). Mathematics as an Educational Task. Dordrecht: Reidel.
Learning Trajectories and Local Instruction Theories 13

Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht:
Reidel.
Freudenthal, H. (1991). Revisiting Mathematics Education. Dordrecht: Kluwer Academic
Publishers.
NCTM Research Advisory Committee (1996). Justification and reform. Journal for Research
in Mathematics Education 27 (5), 516-520.
Simon, M.A. (1995). Reconstructing mathematics pedagogy from a constructivist perpective.
Journal for Research in Mathematics Education, 26, 114-145.
Stephan, M., Bowers, J., Cobb, P., & Gravemeijer, K. (in press). Supporting students’
development of measuring conceptions: Analyzing students’ learning in social context. Journal
for Research of Matematics Education Monograph.
Thompson, A. G., Philipp, R. A., Thompson, P. W., & Boyd, B. (1994). Calculational and
conceptual orientations in teaching mathematics, 1994 Yearbook of the National Council of
Teachers of Mathematics. Reston, VA: National Council of Teachers of Mathematics.
Treffers, A. (1987). Three Dimensions. A Model of Goal and Theory Description in
Mathematics Education. The Wiskobas Project. Dordrecht: Reidel.
Treffers, A. en E. de Moor (1990). Proeve van een nationaal programma voor het reken-
wiskundeonderwijs op de basisschool. Deel II. Basisvaardigheden en cijferen. (A specimen of a
national program for mathematics education in primary school. Part II. Basic skills and written
algorithms).Tilburg: Zwijsen.
Whitney, H. (1988). Mathematical reasoning, early grades. Princeton (Paper).
Learning Trajectories and Local Instruction Theories 14

1 I should be noted that the description of developmental research presented here also builds on what is learned in the
collaboration of the author with Paul Cobb, and Kay McClain at Vanderbilt University (see also Cobb & Gravemeijer,
2001).
2 The retrospective analysis may spark ideas that surpass what is tried out in the classroom. This may create the need for
a new developmental-research project, starting out with a new conjectured local instruction theory. We may further note
that in addition to retrospective analyses that directly aim at reconstruction and revision, the aim of a retrospective
analysis may also be to place classroom events in a broader context by framing them as instances of more encompassing
issues.
3 Actually, there was a dual goal, linear measurement and flexible arithmetic (see for instance Stephan, Bowers, Cobb,
& Gravemeijer, in press); in this article, however, we will limit ourselves to the arithmetic part.
Gravemeijer, K. (2004). Learning Trajectories and Local Instruction Theories as Means of Support
for Teachers in Reform Mathematics Education. Mathematical Thinking and Learning, 6(2).
105-128.