Gravemeijer (2004)
Gravemeijer (2004)
Gravemeijer (2004)
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 depar-
ture 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 addi-
tion 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 the-
ory 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 addi-
tion 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.
INTRODUCTION
In the 1960s and 1970s theories for instructional design were in vogue in the edu-
cational-research community. The most well-known design theories from that pe-
riod are probably Gagné’s Principles of Instructional Design (Gagné & Briggs,
1974). Since then, the interest for instructional design has faded away. More re-
Requests for reprints should be sent to Koeno Gravemeijer, Freudenthal Institute & Department of
Educational Research, Utrecht University, Tiberdreef 4 3561 GG Utrecht, The Netherlands. E-mail:
k.gravemeijer@fi.uu.nl
106 GRAVEMEIJER
How do I create experiences for my students that connect with what they know and
care about but also transcend the present? How do I value their interests and also con-
LOCAL INSTRUCTION THEORIES AS SUPPORT 107
nect them to ideas and traditions growing out of centuries of mathematical explora-
tion and invention? (p. 375)
my view, local instruction theories can never free the teachers from having to de-
sign hypothetical learning trajectories for their own classroom. Nevertheless, I
would argue, that using a local instruction theory as a framework of reference
could enhance the quality of the learning trajectories.
This, in a sense, is the main point of this article: Externally developed local in-
struction theories are indispensable for reform mathematics education. It is unfair
to expect teachers to invent hypothetical learning trajectories without any means of
support. In addition, it can be argued that without them, the chances to reconcile
openness toward students’ own contributions and aiming for given end goals are
very slim.
To develop local instruction theories to support teachers, a theory is needed on
how to help students’ construct mathematical ideas and procedures. The point of
departure here is that RME offers such a theory and that design research is the ap-
propriate method for developing local instruction theories. I elucidate this in the
following way. I start with a description of design research as a method for devel-
oping local instruction theories. Next I use this as a background to elucidate the
RME theoretical framework on the basis of an exemplary local instruction theory. I
complement this by highlighting the very aspects in which the local instruction
theory goes beyond the level of an instructional sequence in terms of a series of in-
structional activities.
DESIGN RESEARCH
Preliminary Design
Design research that focuses on the development of local instruction theories basi-
cally encompasses three phases: developing a preliminary design, conducting a
teaching experiment, and carrying out a retrospective analysis. The first phase
starts with the clarification of mathematical learning goals, combined with antici-
patory thought experiments in which one envisions how the teaching-learning pro-
cess can be realized in the classroom. This first step results in the explicit formula-
tion of a conjectured local instruction theory that is made up of three components:
(a) learning goals for students, (b) planned instructional activities and the tools that
1Accidentally, the same label, developmental research, is used for another Dutch research approach.
Instructional design is also at the heart of this research approach; however, the goal is not to develop do-
main specific instruction theories but to develop improved design theories (van den Akker, 1999).
2It 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 Uni-
versity (see also Gravemeijer & Cobb, 2001).
110 GRAVEMEIJER
will be used, and (c) a conjectured learning process in which one anticipates how
students’ thinking and understanding could evolve when the instructional activities
are used in the classroom.
This conjectured local instruction theory is open to adaptations on the basis of
input of the students and assessments of their actual understandings. The theory
also reflects the importance of anticipating the possible process of their learning as
it could occur when planned instructional activities are used in the classroom. The
manner in which a conjectured local instruction theory is construed can be de-
scribed as “theory-guided bricolage” (Gravemeijer, 1994) because it resembles the
manner of working of an experienced “tinkerer,” or “bricoleur.” The design re-
searcher follows a similar approach using and adapting existing ideas and materi-
als, but the way in which selections and adaptations are made is guided by a the-
ory—in our case, RME theory. RME theory offers three design heuristics, denoted
as guided reinvention, didactical phenomenology, and emergent modeling (which I
discuss in more detail later). These design heuristics help the research team in de-
signing a possible learning route together with a set of potentially useful instruc-
tional activities that fit this learning route. More specifically, this implies that the
researchers think through what mental activities of the students can be expected
when they engage in the instructional activities and how those mental activities can
help the students to develop the envisioned mathematical insights. In the teaching
experiment, those conjectures are put to the test.
framework of Cobb and Yackel (1996) can be used to help the researchers make
sense of classroom events.
Retrospective Analysis
The results of design experiments cannot be linked to pretest and posttest results in
the same direct manner as is common in standard formative evaluation, because
the proposed local instruction theory and prototypical instructional sequence will
differ from what is tried 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 in-
structional 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 se-
quence will have affected the students. Therefore, adaptations will have to be made
when the nonfunctional, or less functional, activities are left out.
Consequently, the instructional sequence is 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 is based on the delib-
erations and the observations of the research team. In this manner, the result of a
developmental research experiment is a well-considered and empirically grounded
rationale for the envisioned learning route in connection with the proposed set of
instructional activities.3 Methodologically, this result has to be justified by the
learning process of the research team. In relation to this, we can refer to a method-
3The retrospective analysis can spark ideas that surpass what is tried out in the classroom. This can cre-
ate the need for a new developmental research project, starting with a new conjectured local instruction
theory. In this manner, subsequent teaching experiments can become part of a series of macro cycles of ex-
perimentation and revision.
LOCAL INSTRUCTION THEORIES AS SUPPORT 113
In the following, I discuss the core elements of a local instruction theory on the ba-
sis of an instructional sequence that is developed in a teaching experiment in Nash-
ville, Tennessee by Cobb, Gravemeijer, McClain, and Stephan of Vanderbilt Uni-
versity (Stephan, Bowers, Cobb, & Gravemeijer, 2000). But, before focusing on
the design of the instructional sequence and the corresponding local instruction
theory, I want to stress the importance of the classroom culture that is essential for
the enactment of such 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 can include expectations and obli-
gations regarding explaining and justifying solutions, attempting to make sense of
explanations given by others, indicating agreement and disagreement, and ques-
tioning alternatives in situations in which a conflict in interpretations or solutions
has become apparent.
In addition to this, certain socio-math norms must be established to create the
opportunity for the students to evaluate mathematical progress.
The Design
The goal of the instructional sequence I use as an example is to foster the use of
flexible mental computation strategies for addition and subtraction up to 100.4 In
designing a conjectured local instruction theory, we can build on the experience
gathered in several decades of developmental/design research at the Freudenthal
Institute and elsewhere. This research effort has resulted in a domain-specific in-
struction theory that is grounded in numerous concrete elaborations of the RME
approach (Gravemeijer, 1994; Streefland, 1990; Treffers, 1987). By interpreting
4Actually, there was a dual goal: linear measurement and flexible arithmetic (e.g., Stephan, Bowers,
Cobb, & Gravemeijer, 2004). In this article, however, I limit myself to the arithmetic part.
114 GRAVEMEIJER
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 10s and 1s.
Beishuizen (1993) found that procedures based on splitting 10s and 1s leads to
more errors than solution procedures that are based on counting on and counting
back. Moreover, the latter type leaves room for a wide variety of solution procedures
and offers more opportunities for curtailing and inventing shortcuts.
Counting-by-jumps therefore fits best the type of instructional sequence we aim for.
It can further be noted that, as the example shows, decuples are used as refer-
ence points in this counting-by-jumps strategy. In relation to this, decuples also
play a central role in framework of number relations that we want the students to
develop.
The RME-guided reinvention heuristic is connected with mathematizing; the
students invent by mathematizing. The idea is that the students not only
mathematize contextual problems—to make them accessible for a mathematical
approach—but also mathematize their own mathematical activity, which brings
their mathematical activity at a higher level. Freudenthal (1971) characterized
mathematizing as a form of organizing, which is also a key element of his
didactical phenomenology (Freudenthal, 1983) that constitutes the second design
heuristic.
Didactical phenomenology is grounded in a phenomenology of mathematics,
within which the focus is on the relation between a mathematical “thought thing”
(nooumenon) and the “phenomenon” it describes and analyses, or, in short, organizes.
The didactical phenomenological analysis can orient the researchers toward ap-
plied problems that can be suitable as points of impact for a process of progressive
mathematization. So, rather than looking around for material that concretizes a
given concept, the didactical phenomenology suggests looking for phenomena that
might create opportunities for the learner to constitute the mental object that is be-
ing mathematized by that very concept.
In relation to a phenomenology of numbers, Freudenthal (1983) noted, “num-
bers organize the phenomenon of quantity,” whereas “the phenomenon ‘number’ is
organized by means of the decimal system” (p. 28). He worked this out in more de-
tail for addition, starting with the lowest level, which is to combine two sets—as in
5 cars and 3 cars or 5 marbles and 3 marbles. However, he argued, problems arise
when the addition is not plainly recognizable as the union of two sets, as is the fol-
lowing case: John has 5 marbles, and Pete has 3 more. How many does Pete have?
Instead of uniting two given sets, the students must consider the imaginary set of
Pete as split into two sets, and reason from there.
Next to those situations, in which addition is not plainly recognizable as the un-
ion of two sets, there are also situations in which it is less natural to speak of sets
consisting of 5 and 3 elements, such as 5 steps (of stairs) and 3 steps, 5 days and 3
days, or 5 kilometer and 3 kilometer. With those spatial or temporal phenomena
one cannot speak of a union of two unstructured sets. Instead, counting is used to
organize magnitudes, in which measuring the magnitude is articulated by the natu-
ral multiples of a unit. Continuous phenomena are made discrete by a one-to-one
mapping of the successive intervals on a sequence of points that follow each other
in space or time, in a process that in turn suggests a counting process. In line with
this sequential character, the results of additions of magnitudes are obtained by
counting on. In relation to this, Freudenthal (1983) pointed to the close relation be-
tween cardinal and ordinal numbers: “5 + 3 is defined cardinally, but from olden
times it has been calculated ordinarily” (p. 99). The result of 5 + 3 is obtained by
starting with the mental 5, and counting on, 6, 7, 8.
From this phenomenological analysis, Freudenthal (1983) concluded,
“Counting can and must immediately be transferred from discrete quantities, repre-
sented by sets, to magnitudes” (p. 101). He recommended the number line as a device
that visualizes magnitudes and, at the same time, the natural numbers. The number
line also lends itself to express more or less as directions. In this manner, the number
line, or two parallel number lines, can also be used to visualize the problem of Pete
LOCAL INSTRUCTION THEORIES AS SUPPORT 117
who has three marbles more than John.5 This reference to the use of the number line
brings us to the issue of models and modeling, for example, the next design heuristic.
The third heuristic, the emergent-modeling design heuristic, assigns a role to
models that differs from the role of ready-made models as embodiments of abstract
concepts mentioned earlier. Instead of trying to concretize abstract mathematical
knowledge, the objective is to try to help students model their own informal mathe-
matical activity. The aim is that the model with which the students model their own
informal mathematical activity gradually develops into a model for more formal
mathematical reasoning. However, the model I am referring to is more an overarch-
ing concept than one specific model. In practice, the model in the emergent modeling
heuristic is actually shaped as a series of consecutive symbolizations or tools6 that
can be described as a cascade of inscriptions or a chain of signification. From a more
global perspective, these tools can be seen as various manifestations of the same
model. So when I speak of a shift in the role of the model in the following, I am talking
about the model on a more general level. On a more detailed level, this transition can
encompass various tools that gradually take on different roles.
The label emergent refers both to the character of the process by which models
emerge within RME and to the process by which these models support the emer-
gence of formal mathematical ways of knowing. According to the emergent-mod-
els design heuristic, the model first comes to the fore as a model of the students’ sit-
uated informal strategies. Then, over time the model gradually takes on a life of its
own. The model becomes an entity in its own right and starts to serve as a model for
more formal, yet personally meaningful, mathematical reasoning. In relation to
this, we discern four different types or levels of activity (Gravemeijer, 1999, 2002):
5Although the design of the instructional sequence under discussion is in line with the previously dis-
cussed elaboration of a didactical phenomenology of number, it should be noted that Freudenthal’s elabo-
ration was not the actual source for the design. The research team came to similar conclusions on the basis
of didactical phenomenological considerations in connection with earlier design experiments. The re-
search team’s didactical-phenomenological deliberations build on the observation that students tend to
come up with a wide variety of counting solutions when confronted with linear-type context problems
(e.g., Vuurmans, 1991). In addition, a closer look at counting strategies shows us 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, whereas the solution procedures consist of
moving up and down the number sequence. We argue that it is important that the students connect the first
and the latter. This then inspired us to try to integrate measurement and the empty number line.
6I use the word “tools” as a generic term in the following discussion, encompassing also symboliza-
tions, or inscriptions.
118 GRAVEMEIJER
These levels of activity underline that the model is grounded in students’ under-
standings of paradigmatic, experientially real task settings. In other words, the
model emerges as situation-specific imagery. This implies that initially, at the ref-
erential level, the model is meaningful for the students because it signifies for them
the activity in the task setting to which it refers. General activity begins to emerge
as the students start to reason about the mathematical relations involved. As a con-
sequence, the model loses its dependency on situation-specific imagery and gradu-
ally develops into a model that derives its meaning from a framework of mathemat-
ical relations that is being construed in the process. The transition from model-of
to model-for coincides with a progression from informal to more formal mathe-
matical reasoning that involves the creation of new mathematical reality, which is
thought of as consisting of mathematical objects (Sfard, 1991) within a framework
of mathematical relations. The level of more formal activity is reached when the
students no longer need the support of models.
As an aside, it can be noted that this transition cannot be pinned down to one
specific symbolization or tool. Instead, there is a gradual change in the way the stu-
dents perceive and use tools as their personal framework grows.
Several authors have proposed the use of the number line as a means of support
for addition and subtraction up to 100 (Freudenthal, 1983; Treffers & de Moor,
1990; Whitney, 1988). For me, the objective to help students make a connection
between the cardinal and the ordinal aspect forms the main 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 count-
ing strategies—by describing the subsequent counting steps as arcs on an empty
number line (Gravemeijer, 1994, 1999). I speak of an empty number line because
this number line is empty except for the numbers that are actually needed. The stu-
dents add these to the number line as a part of the solution process (see Fig. 3).
The interpretation of the number line, however, is not self-evident. For the stu-
dents, it does not speak for itself what the marks on the number line signify. The hash
marks might signify either cardinal or ordinal numbers and not necessarily a synthe-
sis of the two. Exactly for that reason, Whitney (1988) and Treffers (Treffers & de
Moor, 1990) let their introduction of the number line be preceded by activities with a
bead string. This bead string consists of 100 beads, colored in groups of 10. While ex-
ecuting various counting tasks, students find out that the decimal structure can be
used to solve tasks like, “Count 38 and add 24 more. Which number do you get?”
This solution procedure is being modeled with arcs on the empty number line. The
indispensability of this kind of imagery proved itself in a design experiment where
the bead string was skipped (Cobb, Gravemeijer, Yackel, McClain, & Whitenack,
1997). The students did not have the means to resolve among themselves whether a
hash mark with a number, say 38 for example, should be thought of as signifying 38
objects (e.g. candies or beads) or the thirty-eighth object.
I concur with Freudenthal’s (1983) arguments to ground addition and subtrac-
tion in linear measurement, as measuring presents itself as a natural alternative for
counting on a bead string. Conceptual understanding of measurement requires that
students interpret the activity of measuring as the accumulation of distance
(Thompson & Thompson, 1996). Similarly, a number on a ruler would have to sig-
nify the total measure of the distance measured. Speculating on the genesis of the
ruler in history, one can take the view that the ruler came about as a curtailment of
iterating a measurement unit. So, the ruler can be thought of as a model of iterating
some measurement unit, whereas the empty number line can function as a model
for more sophisticated mathematical reasoning in the context of mental computa-
tion strategies with numbers up to 100. The connection between the two can be
made by building on the relation between iterating measurement units as accumu-
lating distances and a cardinal interpretation of positions on the number line. This
is truly a model-of/model-for transition if it coincides with a shift in the student’s
view of numbers as referents of distances to numbers as mathematical entities.
This shift involves a transition from viewing numbers as tied to identifiable objects
or units (i.e., numbers as constituents of magnitudes, such as 38 feet) to viewing
numbers as mathematical objects (e.g., 38). For the student, a number viewed as a
mathematical object still has quantitative meaning, but this meaning is no longer
dependent upon its connection with identifiable distances or with specified count-
able items. Instead, numbers viewed as mathematical objects derive their meaning
from their place in a network of number relations.
The enacted instructional sequence. With the help of the previously dis-
cussed elaboration of the design heuristics, we developed a preliminary design of
the instructional sequence, which was worked out in the teaching experiment in
Nashville, Tennessee, which is well-documented in various publications (e.g.,
Gravemeijer, 1999; Stephan, 1998; Stephan et al., in press; Stephan, Cobb,
Gravemeijer, & Estes, 2001). Space does not allow for an elaborate account of all
findings here. Instead, I give a brief description of the enacted instructional se-
quence, supplemented with elements of the retrospective analysis that offer essen-
tial background theory for teachers. With the latter, I want to highlight the impor-
tance of offering teachers more than a set of instructional activities. Note that I
120 GRAVEMEIJER
7Because Unifix cubes can be clicked together, it is rather easy to make a solid stack of cubes.
TABLE 1
Overview of the Proposed Role of Tools in the Instructional Sequence
Masking tape Record of activity of pacing Reasoning about activity of pacing Focus on covering distance
Footstrip Record of pacing (builds on masking tape) Measuring with a “big step” of five = measuring by Measuring as divorced from activity of measuring
(Form/function shift: using a record of pacing as iterating a collection of paces Structuring distance in collections of 5s and 1s
a tool for measuring)
Smurf cans Stack of Unifix cubes signifies result of iterating Measuring by creating a stack of Unifix cubes Builds on measuring divorced from activity of iterating
Smurf bar Signifies result of iterating Measuring by iterating a collection of 10 Unifix cubes Accumulation of distances
Structuring distance into measures of 10s and 1s Coordinating measuring with 10s with measuring by 1s
10-strip Signifies measuring 10s and 1s with the Smurf Measuring by iterating the 10-strip, and using the Accumulation of distances
bar strip as a ruler for the 1s Coordinating 10s & 1s
Measurement strip Signifies measuring with 10 strip/ (1) Measuring: strip alongside item; counting by 10s and Distance seen as already partitioned; extension
Starts to signify result of measuring 1s already has a measure
(Form/function shift: inscription developed => reading of endpoint Part-whole reasoning/quantifying the gaps between
for measuring is used for scaffolding (2) Reasoning about spatial extensions (results of two or more lengths
and communicating) measuring have become entities in and of Shift in focus: focus on number relations; developing and
themselves) using emergent framework of number relations
Empty number line Signifies reasoning with measurement strip Means of scaffolding & means of communicating Numbers as mathematical entities (numbers derive their
about reasoning about number relations meaning from a framework of number relations)
Various arithmetical strategies
Note. Reprinted with permission from Stephan, M., Bowers, J., Cobb, P., & Gravemeijer, K. (Eds.), Supporting students’development of measuring concep-
121
tions: Analyzing students learning in social context. Journal for Research in Mathematics Education Monograph No. 12. Copyright 2003 by the National Coun-
cil of Teachers of Mathematics. All rights reserved.
122 GRAVEMEIJER
classrooms. The first concerns an empirically grounded theory on how the students
are expected to make sense of acting with new tools and how this relates to preced-
ing activities. The second concerns an empirically grounded theory on the stu-
dents’ conceptual development in relation to the relevant mathematical concepts.
measuring with the footstrip enables the students to consider a stack of Unifix
cubes as a result of measuring, an observation that was underscored by the stu-
dents’ inability to do this when we tried to introduce the Unifix cubes too quickly.
The transition from measuring with individual cubes to measuring with the
Smurf bar is similar to the transition between pacing and measuring with the
footstrip. Next, measuring with the 10-strip builds on the history of measuring with
the Smurf bar. Thanks to this history, measuring with the 10-strip signifies measur-
ing 10s and 1s with the Smurf bar for the students. In turn, measuring with the mea-
surement strip builds on the imagery of measuring with the 10-strip. Actually, the
students initially count by 10s and 1s on the measurement strip to establish the
length of a item. Gradually, however, a position on the measurement strip starts to
signify the result of measuring. Then, we meet another form/function shift, when
the tasks shift from measuring to reasoning about measures, and the measurement
strip is used as a means for scaffolding and communicating ways of reasoning.
Finally, drawing arcs on the empty number line is introduced as an alternative
means for scaffolding and communicating ways of reasoning with the measure-
ment strip.
The main idea behind the design of this cascade of tools is that the way in which
the students act and reason with each tool builds on their activity with earlier ones.
This build up is to ensure that the students have a meaningful way of acting with
the tools, because they can rely on the imagery of acting with earlier, already famil-
iar tools. From this perspective, it is essential that the teachers who want to reenact
the sequence come to grips with the researchers’ empirically grounded theory on
how reasoning with one tool builds upon the other.
steps it took them to reach the end of the item. This was inferred from the fact that
some students did not count the placement of the first foot (when the heel was aligned
with the beginning of the item measured). They started counting, “one,” with the
placement of the second foot, whereas others started their counting with the first
foot. What we are aiming for is that the students come to see measuring as covering
amounts of space. To reach this goal, the teacher has to make the two different ways
of measuring a topic of discussion (whether you have to count the first foot). In such a
discussion, the students can start to realize that it is not just a matter of convention; in-
stead, if one does not count the first foot, an amount of the item would not be mea-
sured. In this manner, students can come to see the goal of measuring as covering
amounts of space—as was the case in the experimental classroom.
With the activity of measuring with a footstrip of five paces, many students ran
into problems when the space that was being covered by the footstrip extended past
the physical extension of the measured item. For them, apparently, measuring was
tied to the physical act of placing a footstrip, and they could not mentally cut the
footstrip when needed. The instructional goal here is that the space to be measured
takes priority over the measurement activity and becomes independent of activity for
the students. Whole-class discussions on concrete instances are needed to create op-
portunities for students to articulate how the extended footstrip can be mentally cut.
The activity of measuring with a Smurf bar in the Smurf scenario showed that,
for some students, the curtailment of counting by individual cubes was based only
on a number word relation. For instance, when the second iteration of the Smurf
bar would extend beyond the item measured, they would count the cubes past the
first iteration as “21, 22, 23 and so forth”, instead of, “11, 12, 23,.…” For them,
“20” seemed to be the number word associated with the second placement of the
Smurf bar rather than the amount of space covered by 20 cubes. Although what is
aimed for is that the students realize that as “20” signifies the length covered by 20
cubes, 21, 22, and 23 must extend beyond the length whose measure was 20. In
other words, the students have to come to grips with coordinating measuring with
10s with measuring with 1s. To achieve this, the latter has to become a topic of dis-
cussion in the classroom.
In subsequent activities, the Smurf bar is replaced by paper 10-strips and next
ten 10-strips are taped together to make a measurement strip of 100 cans long.
Here, the students may initially measure with the measurement strip by laying the
strip down alongside the item and counting by 10s and 1s until they reach the end-
point of the item. Gradually, however, the students curtail their activity of counting
up on the measurement strip and find that the length of an item can be measured by
laying down the measurement strip alongside the item and simply reading off the
numeral corresponding to the position of the farthest endpoint. To do this
insightfully, they have to conceive an extension as already having a measure, inde-
pendent of the activity of measuring.
The next set of activities involves tasks such as comparing the lengths of two
items and figuring out the difference with the help of the measurement strip. This is
LOCAL INSTRUCTION THEORIES AS SUPPORT 125
the first instructional activity in which the students do not measure an item that is
physically present. The mathematically significant issue here is the quantification of
a gap between two numbers. In the teaching experiment, some students counted the
spaces between the two numbers whereas others counted the lines. To overcome this
problem, the teacher made the different ways of quantifying the gaps a topic of dis-
cussion and asked the students to explain what each line or space signified to them.
As a result of such discussions, the method of counting spaces to specify the measure
of the spatial extension between two lengths became taken-as-shared.
A next step that is aimed for is that the students gradually replace the method of
literally counting spaces by arithmetical reasoning. Again the teacher plays an im-
portant role by stimulating discussions on the different ways of establishing the
number of spaces.
Finally, the empty number line is introduced as a means of describing and scaf-
folding various forms of arithmetical reasoning. When making the transition from
the measurement strip to the number line, it is essential that the students differenti-
ate between the activity of measuring and the activity of representing arithmetical
strategies. On the empty number line the goal is for students to express how they
would, for instance, increment 64 with 28. For example, by first measuring 64,
then adding six 1s, which would get one to 70, then measuring two times 10, which
would result in 80 and 90, respectively, and finally adding two 1s, which adds up to
92. When describing this method, it would be sufficient to show that when starting
at 64, add 6, arrive at 70; then add 10, arriving at 80; another 10, arriving at 90; and
2, arriving at 92. To strive for an exact proportional representation of all the jumps
would severely hamper a flexible use of the number line. Therefore we must make
sure that the students are aware of the distinction between the ruler as a measure-
ment 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. As argued before, this shift involves a transition
from viewing numbers as tied to identifiable objects or units to viewing numbers as
entities on their own that derive their meaning from a framework of number rela-
tions. This framework of number relations, then, offers the basis for flexible men-
tal computation strategies for addition and subtraction up to 100, which was our in-
structional goal.
CONCLUSION
The main issue of this article is what instructional design has to offer to reform
mathematics education, whereas classical instructional design theories do not fit
126 GRAVEMEIJER
mathematics education that tries to capitalize on the inventions of the students. The
classical approach of task analysis results in a breakdown of the mathematical con-
tent in a hierarchy of small learning objectives that have to be mastered in a fixed
sequence. This sequence is to be followed independent of the input or interest of
the students; the only variation is one in speed and reteaching. A final drawback of
the analytically defined learning objectives is that the students cannot see the rele-
vance until they have reached the end of the process.
Still, I argue, if justice is to be given to the input of the students and their ideas
built on, a well-founded plan is needed. In this respect, I point to the proactive role
of the teacher in establishing an appropriate classroom culture, in choosing and in-
troducing instructional tasks, organizing group work, framing topics for discus-
sion, and orchestrating discussion. Following Simon (1995), this implies design-
ing, enacting, assessing, and revising hypothetical learning trajectories in an
iterative series of mathematical teaching cycles.
I use the example of the local instruction theory on addition and subtraction to
show that design research can help teachers by developing viable local instruction
theories, which can be used by classroom teachers to construe hypothetical learn-
ing trajectories that fit the characteristics and actual situations of their own class-
rooms. I highlight the word theory because, in contrast with traditional design the-
ories, the emphasis is not on an elaborated instructional sequence with detailed
directions for the teacher, but on the theory that underpins a possible instructional
sequence—a theory of which we claim offers an empirically grounded theory on
how the instructional sequence can work. The examples of the theory behind the
way the various tools build on each other and the theory on how the conceptual de-
velopment of the students can be supported by exploiting potential mathematical
topics for discussion shed light on the theoretical framework that teachers need to
make informed decisions in the classroom. In line with the RME theory that in-
spired the design, this enables teachers to design instruction that helps students to
develop their current ways of reasoning into more sophisticated ways of mathe-
matical reasoning.
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