Download Modeling proof competency

Kristina Reiss and Aiso Heinze, University of Munich (Germany)1
Research Paradigms on Teaching and Learning Proof
Modeling proof competency
Summary
In the following, we describe the crucial aspects of our research program on proof and
argumentation. This research focuses on geometrical proof competencies of students at the
lower secondary level (grades 7–9). We have already formulated a preliminary model of
geometrical proof and intend to develop a more detailed competency model on proof that
describes
•
the structure of the individual’s competency when creating proofs, and
•
the development of proof competency including continuous learning processes as well as
sudden insights.
The competency model will be based on theoretical assumptions and has to be confirmed by
empirical data. The model can be applied for a more profound development of proof items in
assessments at the system level (i.e., classrooms or larger units). Moreover, it should help
teachers and researchers to develop curricula and teaching material and to evaluate learning
environments for the mathematics classroom.
Our research is clearly embedded in mathematics education and performed in the mathematics
classroom with a specific emphasize on proof and argumentation. However, it is strongly
interdisciplinary in its nature. We particularly take into account cognitive as well as
motivational aspects of educational psychology and cooperate with researchers from this
field.
1. Proof competency
In the center of our research paradigm is a specific conception of proof competency.
According to Weinert (2001), competencies are defined as cognitive abilities and skills, which
individuals have or which can be learned by them. These abilities and skills enable them to
solve particular problems and encompass the motivational, volitional, and social readiness and
capacity to utilize the solutions successfully and responsibly in variable situations. However,
competencies cannot be seen independent of domains, in particular, in the case of competency
acquisition. In this sense, a description of proof competency is fundamentally based on the
domain of mathematical proof and proving. More precisely, proof competency encompasses
cognitive and non-cognitive personal dispositions as prerequisites for successfully dealing
with problem situations related to mathematical proof.
Based on these assumptions the concept of proof certainly is important for the definition of
proof competency. Our research interest is not to describe the students’ understanding of
mathematical proof explicitly but to describe their competency to solve proof problems.
Therefore we take a normative perspective on mathematical proof. This view is supported by
1
Department of Mathematics, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 München,
Germany, [email protected] .
the curriculum for secondary mathematics classroom and also provides criteria for the
acceptance of students’ proof as mathematically correct.
2. Structure of proof competency
One important application of the concept of “proof competency” probably is to assess whether
an individual is more competent in solving proof problems than another individual. For this
purpose we have to find out what the structure of proof competency is. Therefore it is
important to analyze proof competency with respect to the factors by which it is influenced (in
the sense of subsidiary competencies). Since proof competency is characterized by the
competency to solve proof problems, it goes without saying to start with a theoretical
cognitively oriented investigation of proving processes. This means, that in addition to the
normative perspective on proof there is a second analytical perspective on the process of
generating a proof. An analysis of the proving process is necessary for identifying essential
competencies that are part of the proof competency.
A starting point might be an expert model of the proving process as suggested by Boero
(1999). He distinguished six phases: (1) exploration of the problem situation, (2) formulation
of a hypothesis, (3) exploration of the hypothesis (including the identification of possible
arguments), (4) constructon of a logical chain of arguments, (5) extension of this chain of
arguments to a proof (according to given standards), and optionally (6) approaching a formal
proof. Based on such a process model one may identify knowledge, abilities and skills related
to mathematics, which are necessarily part of proof competency. However, the expert’s steps
while performing a proof do not give detailed information on the process. Accordingly,
aspects of the process should be taken into account. According to the problem-solving
competencies formulated by Schoenfeld (1992) it is possible to describe cognitive factors like
knowledge on facts, strategic knowledge, meta-cognition and non-cognitive factors like
beliefs and motivation as important subelements of proof competency. These factors have to
be analyzed in detail (e.g. Weber, 2001, for aspects of the strategic knowledge). Moreover,
there is a competency on a meta-level concerning the knowledge on characteristics of
deductive reasoning in mathematical proofs and the competency to perform multiple-step
deductive reasoning. The latter includes the ability to understand that statements can have
different status (as premise and as conclusion) and to use this fact for creating deductive
chains of arguments (cf. Duval, 1991). We will call this the “dynamic status” of statements.
Regarding the classification of different epistemologies of mathematical proof by Balacheff
(2002) this view is connected to the one of Healy and Hoyles (1998). Balacheff (2002)
describes this view as “related to the technical understanding of what it [proof] is in the
mathematical activity”. On a statistical level one can find evidence that factors identified
theoretically are related to proof competency. A quantitative analysis of empirical data like
linear regression models can provide insight about the significance of different factors (see the
example below). It is important to notice that research in this paradigm aims at explaining
important factors of individual proof competency. However, it is not possible and we do not
have the ambition to completely explain individual proof competency. Human beings are too
different that their behavior and their actions might be fully explained by a small set of
factors. However, quantitative methods as used in our research (see examples below) might be
appropriate for providing evidence at the system level of a school or a classroom.
Example 1:
A study on geometrical proof competency was conducted with 576 grade 9 students (350
from Germany and 226 from Taiwan). Data was collected by different tests for (1) proof
competency in geometry, (2) competency on calculating angles sizes and side lengths in
geometric figures, (3) knowledge on geometrical concepts, and (4) problem-solving
competency regarding unfamiliar problems related to mathematics. A linear regression
analysis showed that there was a significant influence of knowledge on geometric concepts,
competency in performing geometric calculations and problem solving competency on the
proof competency in geometry. This result differed between German and Taiwanese students.
For German students, the three influential factors explained 32.5% of variance of the proof
competence, for the Taiwanese sample even 57.7% (recent results of a joint project with
Ying-Hao Cheng, Fou-Lai Lin, and Stefan Ufer).
Example 2:
In a three-year longitudinal study on geometrical proof with 194 German students from grade
7 to 9 data were collected each year for (1) proof competency in geometry, (2) competency on
calculating angles sizes and side lengths in geometric figures and, only in grade 9, knowledge
of geometrical concepts. A linear regression analysis to explain proof competency in grade 9
showed that only proof competency in grades 7 and 8 and the knowledge on concepts in grade
9 had significant influence (46% explained variance). Particularly, the competency of
calculate angles and sides in geometrical figures in grade 7, 8 and 9 had no significant
influence.
3. Modeling proof competency
Factors which influence proof competency can be identified by a theoretical analysis of
proving activities and evaluated by empirical findings. They have to be integrated into a
competency model. The naive approach of putting together all the important factors is not
consistent with the idea of modeling a competency. In particular, the aim to use a competency
model for the description of the development of proof competency implies that the
preparation and validation of such a model is a complex challenge.
Since proof competency is described by the competency of an individual to solve proof
problems, the main point is to identify characterizations of the difficulty of proof problems. In
the following an approach for a competency model on proof is given for geometric proof
competency of students in grades 7–9.
Based on the elaboration of the previous section we can assume that
•
creating a proof consisting of one deductive step is more difficult than solving a simple
geometrical calculation problem, because the competency of deductive reasoning
combined with explicit conceptual knowledge is necessary for the first task but not for the
second. Solving simple geometrical calculation problems means to apply rules in concrete
situations. The underlying principles do not have to be given explicitly, because for the
solutions reasoning is not required. This means that one-step reasoning presupposes a
different kind of competency.
•
creating a proof consisting of multiple deductive steps is more difficult than creating a
proof consisting of one deductive step, because combining multiple steps requires
strategic knowledge as well as the understanding and application of the dynamic status of
statements.
•
distinguishing between two-step proofs, three-step proofs etc. does not seem to be
essential. In fact, there is a general tendency of an increase in complexity and difficulty
going along with the number of steps of a proof. However, it is assumed that there is no
new level of quality in the sense of a non-continuous increase of complexity.
This first analysis revealed three levels of competency:
Level 1
applying rules in
concrete situations
Level 2
performing single
step proofs
Level 3
performing multiple
step proofs
Although the competency model was developed in a research program on geometric proof,
the description of the competency levels are quite general and independent of the actual
content. If we apply this model to geometry, for example in order to develop test items, it
turns out that it is not sufficient to consider the domain-specific content only from a
mathematical perspective. The mathematical perspective gives a first orientation about the
structure of a proof, the concepts involved and the number of deductive steps. However, with
respect to the competency model it remains to be shown that during the cognitive process the
individual really performs all these steps separately. Particularly at this point it is evident, that
this type of research is based on a research paradigm integrating aspects of mathematics
education and of educational psychology.
We assume that the domain-specific content plays two different roles. On the one hand, the
difficulty of proof problems will increase with the complexity of the concepts involved. For
example, a proof step using the vertical angles property is less difficult than a proof step based
on the congruence rule SAS, because for the latter it is more complex to check whether the
conditions of the rule are fulfilled. On the other hand, the quality of the individual conceptual
knowledge plays a role. There may be an individual who retrieves a “schema-congruent rule
SAS” including all conditions and consequences from his or her mind (in the sense of a
chunk, cf. Anderson, 2004). And there may be an individual who needs a lot of cognitive
capacity to reconstruct the congruence rule before being able to apply it. The former is
probably more successful with multiple step proofs including the SAS rule than the latter. The
quality of knowledge may be a key factor when classifying proof problems as one step or
multiple step proofs. Mathematically, a proof may consist of two steps (competency level 3),
but for an individual solving the proof problem the first proof step probably is “only” initial
information in its mental representation of the problem, because he or she just retrieves this
information from memory. In this case the proof problem requires only the competency of
level 2 (performing a one step proof). This means that the same proof problem may be
allocated in different competency levels for different individuals. In this sense, we
hypothesize that the quality of individual knowledge has a moderating effect in the relation
between proof problems and competency levels.
Summarizing the previous aspects we may state: when applying the competency model we
have to make assumptions about the geometrical knowledge of the sample. Generally, we will
do this in a normative way based on the curriculum. Thus we can develop test items for proof
tests for different grades reflecting the model of proof competency. Actually the same proof
item may be allocated to competency level 3 for the test in grade 7 and to competency level 2
for the test in grade 9.
Empirical results from different studies in Germany (N > 2000) have confirmed the
competency model, particularly the organization of geometrical proof competency in levels
(e.g. Reiss, Hellmich & Reiss, 2002). Data of German students from grade 7 through 9 have
shown repeatedly that for each grade the corresponding test items could be modeled onedimensionally by the dichotomous Rasch model (Rasch, 1960). Recently this model was also
confirmed in a cross-cultural study with German and Taiwanese students for grade 9 (cf. the
study in example 1). The results are given in the following table:
Percentages of
correct solutions
grade 9
Taiwan
lower
third
average
third
upper
third
Germany lower
third
average
third
upper
third
Level 1
applying rules in
concrete situations
Level 2
performing single
step proofs
Level 3
performing multiple
step proofs
54,4
25,0
6,2
69,3
63,6
39,7
76,0
83,2
84,3
60,2
15,7
2,9
83,2
39,6
8,1
88,9
70,0
33,7
4. Open questions and further research
There are several possibilities to extend the present research.
a) We need a deeper analysis of and more empirical evidence regarding the role of the
quality of knowledge and its role as a moderator between the objective mathematical
character of proof problems and the required individual competency when solving this
problem. Here, a test concerning the quality of the individual knowledge should be
developed. In particular, we have to determine which aspects of knowledge will play a
role (only concepts or also knowledge on domain-specific strategies like the examples in
Weber, 2001).
b) It is an open question whether three levels of competency provide enough detailed
information. It is possible that level 2 (one step proofs) can be differentiated in two further
sublevels based on the complexity of the concepts involved (e.g., vertical angle argument
versus congruence rule). Moreover, it is possible that there is an intermediate level
between levels 1 and 2 consisting of geometrical calculation items that require complex
problem-solving strategies (Hsu, 2007, used such items for a qualitative study).
c) Presently, the competency model is restricted and evaluated for geometry. It should also
be applied or adapted for algebra, calculus, and perhaps statistics/stochastics. First
attempts may easily be conducted by re-analyses of existing data.
d) Our experimental studies include questionnaires on interest and motivation. There is some
evidence that interest and motivation play a significant role with respect to proving
competency. Accordingly, these constructs should better be integrated into further
research and dealt with in more detail.
References
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