# Gravemeijer (2004)

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

The article *Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education* was written by Koeno Gravemeijer and published in *Mathematical Thinking and Learning* in 2004. The article is available from Taylor & Francis Online at http://www.tandfonline.com/doi/abs/10.1207/s15327833mtl0602_3.

## 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.

## Outline

The headings in the article are structured as follows:

- Introduction
- Design Research
- Preliminary Design
- The Teaching Experiment
- Retrospective Analysis

- Exemplary Local Instruction Theory
- The Design
- The enacted instructional sequence
- The local instruction theory

- Tools and Imagery
- Potential Mathematical Discourse Topics

- The Design
- Conclusion

## Summary

Gravemeijer claims in this article that theories for instructional design are making a resurgence after constructivist theories shifted focus to student learning. Unlike instructional design principles of the 1960s and 70s, which were built upon notions of expert knowledge, strategies, and the step-by-step prerequisite skills needed to become expert, Gravemeijer claims mathematics education needs instructional design theories that use student understanding as the starting point and supports student development of more sophisticated ways of reasoning. Gravemeier admits there is a tension between a focus on student construction and well-defined endpoints (Ball, 1993), which leaves teachers to help students construct mathematical knowledge in particular ways (Cobb, 1996). As described by Simon (1995), teachers need *hypothetical learning trajectories* against which they can continually evaluate students' predicted versus observed learning, followed by informed adjustments to their teaching or the hypothetical trajectory. This paper by Gravemeijer addresses the "informed adjustments" and the supports that can be provided to teachers:

The example Simon (1995) worked out shows that designing hypothetical learning trajectories for reform mathematics is no easy task. We can, therefore, ask ourselves what kind of support can be given to teachers. It is clear that we cannot rely on fixed, ready-made, instructional sequences, because the teacher will continuously have to adapt to the actual thinking and learning of his or her students. Thus it seems more adequate to offer the teacher some framework of reference, and a set of exemplary instructional activities that can be used as a source of inspiration. (p. 107)

Gravemeijer introduces the concept of a *local instruction theory* which he describes as "the description of, and rationale for, the envisioned learning route as it relates to a set of instructional activities for a specific topic" (p. 107). A local instruction theory differs from a hypothetical learning trajectory in how it relates to instructional activities. The instruction theories act as somewhat generalized principles and rationales to inform the selection and use of instructional activities, while hypothetical learning trajectories are used in day-to-day planning and reflecting situated in real classroom contexts. By addressing the relationship of learning to the instructional activities, Gravemeijer uses local instruction theories to describe a common foundation teachers can use for building trajectories, saying that "Externally developed local instruction theories are indispensable for reform mathematics education" and that it is "unfair to expect teachers to invent hypothetical learning trajectories without any means of support" (p. 108). Given Gravemeijer's long association with the Freudenthal Institute, he describes how design principles from Realistic Mathematics Education (RME) provide the kind of instructional design framework for creating a local instruction theory.

### Design Research and RME

Developmental (Gravemeijer, 1994, 1998) and design research (Gravemeijer & Cobb, 2001) consists of cyclical iterations of thought experiments, teaching experiments, and retrospective analyses. Design research can be similar to how teachers improve their instruction as they gain experience: they plan an activity for year one, then conduct that activity, then reflect on the activity so it will be better in year two. A team of researchers who carefully theorize, observe, collect data, and analyze the results across multiple classrooms can more quickly and effectively improve tasks and instruction than a teacher can alone.

Gravemeijer describes the design research he conducted with Paul Cobb and others around the development of mental computation strategies for addition and subtraction with elementary students (Stephan, Bowers, Cobb, & Gravemeijer, 2000). The research resulted in a domain-specific instruction theory grounded in three design principles of Realistic Mathematics Education (Gravemeijer, 1994; Streefland, 1990; Treffers, 1987).

#### Guided Reinvention

Freudenthal (1973) believed mathematics is best learned when students get to experience a process of learning that's similar to the way the mathematics was invented.

If mathematics is to be applied, applying mathematics should be taught and learned. Applying is often interpreted, as mentioned above, as substituting numerical values for parameters in general theorems and theories. This is a misleading terminology. Mathematics is applied by creating it anew each time — I will expound this in more detail too. This activity can never be exercised by learning mathematics as a ready-made product. Drilling algorithms may be indispensable, but inventing problems to drill algorithms does not create opportunities to teach applying mathematics. This so-called applied mathematics lacks the flexibility of good mathematics. (Freudenthal, 1973, p. 118)

The goal of guided reinvention is not to replicate the invention of the mathematics, but learn from history how a mathematical idea might be constructed in the mind of a student.

#### Didactical Phenomenology

The concept of didactical phenomenology relates the mathematical "thought thing" and the phenomenon it describes. As described by Freudenthal:

Mathematical concepts, structures, and ideas serve to organise phenomena — phenomena from the concrete world as well as from mathematics — and in the past I have illustrated this by many examples. By means of geometrical figures like triangle, parallelogram, rhombus, or square, one succeeds in organising the world of contour phenomena; numbers organise the phenomenon of quantity. On a higher level the phenomenon of geometrical figure is organised by means of geometrical constructions and proofs, the phenomenon "number" is organised by means of the decimal system. So it goes in mathematics up to the highest levels: continuing abstraction brings similar looking mathematical phenomena under one concept — group, field, topological space, deduction, induction, and so on. (Freudenthal, 1983, p. 28)

Traditionally an abstract mathematics is taught prior to finding examples for students to make the mathematics concrete. With didactical phenomenology, the focus is on progressive mathematization, suggesting "looking for phenomena that might create opportunities for the learner to constitute the mental object that is being mathematized" (Gravemeijer, p. 116).

#### Emergent Modeling

Emergent modeling can be described with an example. Imagine an elementary class learning about fractions. Instead of giving students a formal model (like a numerator and denominator), the concept of emergent modeling says we should let students reach these models informally and progressively. If a task involves the sharing of parts of cookies with the students, students might begin with breaking apart actual cookies. Once realizing this isn't convenient, students might move to drawing cookies on paper. At some point they'll realize that drawing all the details of the cookie isn't necessary and just use a circle to represent a cookie. Up until this point, these are all *models-of* a cookie. The key step in this process is when students start using circles to model other contextual situations, like working with fractions of time, money, space, etc. Now the circle is a *model-for* a part-whole relationship, and not representing a specific object like a cookie. These models-for have the power to generalize to other contexts, and eventually students no longer need the circle and rely on formal mathematics to represent and work with fractions. Gravemeijer describes a similar process in this paper, except with how bead strings, unifix cubes, and rulers can lead to marked and empty number lines as students develop ideas of cardinality, ordinality, and distance as they learn mental strategies for addition and subtraction.

### Conclusion

The three RME principles above — guided reinvention, didactical phenomenology, and emergent modeling — do not describe a detailed instructional sequence of tasks and instructions for a teacher. They are, however, a way of theorizing how a particular instructional sequence should work, grounded in the design research conducted by Gravemeijer et al. This kind of local instruction theory is what allows teachers to design hypothetical learning trajectories that focus on the construction of student understanding, and provide some common ground for helping teachers become better at trajectory hypothesizing.

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### APA

Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. *Mathematical Thinking and Learning*, 6(2), 105–128. doi:10.1207/s15327833mtl0602_3

### BibTeX

@article{Gravemeijer2004, 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.}, author = {Gravemeijer, Koeno}, doi = {10.1207/s15327833mtl0602\_3}, journal = {Mathematical Thinking and Learning}, number = {2}, pages = {105--128}, title = {{Local instruction theories as means of support for teachers in reform mathematics education}}, url = {http://www.tandfonline.com/doi/abs/10.1207/s15327833mtl0602\_3}, volume = {6}, year = {2004} }