Guided Reinvention
A heuristic of the instructional
design theory of Realistic
Mathematics Education (RME)
What is RME?
• Realistic Mathematics Education (RME) is an
instructional design theory which centers
around the view of mathematics as a human
activity (Freudenthal 1991)
• “The idea is to allow learners to come to regard
the knowledge that they acquire as their own
private knowledge, knowledge for which they
themselves are responsible.”(Gravemeijer 1999)
Why RME?
• Most mathematics courses are taught by presenting the
mathematics as a finished product; traditionally in a
definition-theorem-proof (DTP style), and then perhaps
some examples
• In other words, most courses are taught by stating the
general and then moving to more specific ideas to
demonstrate the point
• Freudenthal, one of the pioneers of RME, argues that
mathematics tends to develop the opposite way in the
minds of individuals, from the specific to the general
case
• Lakatos, in his 1976 book Proofs and Refutations, claims
that a finished product, DTP teaching style masks the
process by which the material was discovered
Three Tenets of RME
• Cobb (2000) gives three central tenets of
Realistic Mathematics Education:
▫ Starting points of the instructional sequence
should be ‘experientially real’ to the students
(hence the name realistic mathematics education)
▫ Starting points should be justifiable in terms of
the potential ending points of the sequence
▫ Instructional sequences should contain activities
in which students create and elaborate symbolic
models of their informal activity
Guided Reinvention Projects
• Primary goal: To develop a local (i.e. domainspecific) instructional theory that will allow students
to “[invent] the mathematics themselves” (Larsen
2004). This involves a two-part process:
▫ Step 1: “students are engaged in activities designed to
invoke powerful informal understandings” (Weber &
Larsen, 2008)
▫ Step 2: “students are engaged in activities designed to
support reflection on these informal notions in order
to promote the development of formal concepts”
(Weber & Larsen, 2008)
 this is part of the process of mathematization, defined on
the next slide
Emergent Models
• Emergent models are used in reinvention projects to
support the development of formal mathematical
knowledge from a student’s informal knowledge
(Gravemeijer 1999)
• Mathematization, the mathematical organization of
ideas, is one such way that a student’s informal
knowledge may be adapted into formal mathematics
• Model of / model for: at first, the model should
emerge as a model of the activity of the student in an
experientially real situation, and then evolve into a
model for more formal mathematical activity
(Gravemeijer 1999)
▫ think: referential activity to general activity (Larsen
2004)
Proofs and Refutations as a
framework for reinvention
• Larsen and Zandieh (2007): “Proofs and Refutations
in the Undergraduate Mathematics Classroom”
• Idea: connect the ideas of Lakatos (1976) with the
idea of guided reinvention set forth in Freudenthal
(1971)
▫ Methods of mathematical discovery described by
Lakatos:
 Monster-barring: reject counterexamples as invalid
 Exception-barring: modifying the conjecture to exclude
counterexamples
 proofs and refutations/proof analysis: modify the
conjecture so that the ventured proof “works”
Some Noteworthy Reinvention Projects
Abstract Algebra
• Larsen (2004): students reinvent the concepts of
group and group isomorphism
• Larsen (2010): students reinvent the concepts of
coset and quotient group
Analysis
• Martin (2010): students reinvent the formal
definitions of series and pointwise convergence
A Primary Criticism of Guided Reinvention
• Impractical in the classroom setting to have such a
favorable teacher to student ratio and to devote so
much time to these tasks
▫ Addressed in Larsen (2009): “I must acknowledge
that it is in general not practical in a group theory
course to provide nearly 14 hours of instruction, with a
one-to-two teacher-student ratio, simply to arrive at
the definitions of group and isomorphism.
Nevertheless, this kind of in-depth work with a small
number of students can support the development of an
instructional theory that can in turn be used to design
curriculum that can be practically implemented in an
undergraduate classroom.”
Specific Advantages of Larsen’s
Reinvention of Group and Group
Isomorphism
• Larsen’s 2004 study actualizes several recommendations
from the literature on learning abstract algebra:
▫ Iannone and Nardi (2002) detail the many errors to which
students are prone when ignoring the binary
operation/group structure
▫ Weber (2004) reported that many undergraduates possess
the syntactic knowledge needed to construct proofs, but
lack the strategic knowledge
 Example: his students first attempted to define a bijection
when proving two groups were isomorphic instead of focusing
on the more important “structure-preservation” construct
▫ Leron (1995) suggested developing an intuitive notion of
isomorphism before revealing the formal definition
Why I am interested in reinvention
• My favorite math course of all time – high school
geometry – was taught in a reinvention-type
format
• When I started graduate school, I struggled in the
qualifier courses (taught in DTP format) because I
didn’t have the intuitive feel for the material
• Because I like teaching so much, the idea of
working closely with a small number of students
appeals to me
• Reinvention is much closer to how actual
mathematicians do research