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Proofs in German Mathematics Classroom -
Analysis of Videotaped Lessons - Aiso
Heinze University of Augsburg
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• In the last ten years in mathematics educational
  research several video studies (grade 8/9) were
  conducted, e.g.
• TIMSS Video Study (1997),
• TIMSS-R Video Study (1999),
• Augsburg Video Study (2002),
• Swiss/German Pythagoras Study (2003/4).

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• Aims of these video studies are (among others)
• to get an overview on everyday mathematics
  instruction,
• to analyze mathematics instruction,
• (e.g., macro/micro level)
• to compare mathematics instruction in different
  countries.

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• Analyzing mathematics instruction
• from a mathematical perspective,
• (fundamental ideas, modelling, algorithms ...)
• from a mathematic didactical perspective,
• (preparation of content, working on problems,
  mental models ...)
• from an educational perspective.
• (class/seatwork, teaching style, participation...)

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Some results of TIMSS-Video 1997 (Stigler et al.,
1999)

• Typical Japanese mathematics lesson (50 lessons)
• teacher poses a challenging problem, students
  work at their seats to generate a solution
  (individually/groups),
• during seatwork teacher circulates around the
  class, noting different methods that students are
  constructing,
• after seatwork particular students come to the
  front and share their methods,
• occasionally, teacher provides a brief lecture
  about particular methods that have been shared.
• the cycle teacher presenting a problem, students
  working at their seats, and students sharing
  their solutions with the class, is repeated
  several times during the lesson.

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Some results of TIMSS-Video 1997 (Stigler et al.,
1999)

• Typical German mathematics lesson (100 lessons)
• teacher presents a task such as finding the
  solution set to two simultaneous linear equations
  in two unknowns.
• if it is a new problem teacher works the problem
  at the board, eliciting ideas and procedures from
  the class as work on the problem progresses.
• if the type of problem is already known, a
  student might be called to the chalkboard to work
  the problem the class is expected to monitor the
  students work, to catch errors that are made,
  and to help the student,

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Some results of TIMSS-Video 1997 (Stigler et al.,
1999)

• Typical German mathematics lesson (100 lessons)
• both cases teacher keeps the class moving
  forward by asking questions about next steps and
  about why such steps are appropriate,
• after two or three similar problems have been
  worked, the teacher summarizes the activity by
  pointing to the principle or property that guides
  the deployment of the procedure in these new
  situations,
• for the remaining minutes of the class period,
  teacher assigns several problems in which
  students practice the procedure in similar
  situations.

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Some results of TIMSS-Video 1995

• Typical German teaching style (classwork)
• Students do not work on a complex problem on
  their own the complex problems are solved step
  by step in a classroom discourse which is
  strictly directed by the teacher. The expected
  students answers are only on an elementary
  level.
• In this way a complex open problem is transformed
  into a series of closed, simple questions.
• (Klieme, Schümer Knoll, 2001)

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Video study Augsburg 2002 on proof instruction
Our video study is embedded in the DFG-project
Reasoning and Proof in Geometry. We
investigate how the ability to solve proof
problems develops in the first years of proof
instruction (grade 7/8 in Germany).
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Aims of the DFG-project

• Identifying cognitive and non-cognitive
  individual variables which influence
  argumentation and proof competence,
• Identifying aspects of the mathematics classroom
  which enhance argumentation and proof competence,
• Describing interdependencies between individual
  variables and classroom characteristics.

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Sample (1st period) and Method

• 659 (507) students of grade 7/8
• (358 female and 301 male) in 27 classrooms
• Pretest and posttest mathematics achievement
  (in between regular instruction on proof)
• Questionnaires on argumentation skills,
  scientific reasoning, interest and
  motivation, and mathematical beliefs
• Videotaped classroom observation
• Students questionnaires on classroom instruction

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Video study Augsburg 2002 on proof instruction

• Our video study is based
• on 22 lessons about proofs in geometry,
• grade 8, Gymnasium (high-attaining),
• 8 different classes in 4 schools,
• 2-4 consecutive lessons per class.

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Video study Augsburg 2002 on proof instruction

• Technical information
• two cameras (teacher, class),
• three microphones (one teacher, two class),
• teacher were volunteers,
• students questionnaires as confirmation.

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Video study Augsburg 2002 on proof instruction

• Investigation from a mathematical perspective
• Which components of the mathematical proving
  process can be identified in the videotaped
  proofs?
• Which aspects of proof are emphasized by the
  teachers in the proving process?
• Are there gaps in the proving process or aspects
  that are underemphasized?

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Model of the proving process
1. Production of a conjecture exploration of the
problem situation, identification of arguments
for its plausibility 2. Formulation of the
statement according to shared textual
conventions 3. Exploration of the content of
the conjecture identification of appropriate
arguments generation of a proof idea 4.
Selection and enchaining of coherent, theoretical
arguments into a deductive chain 5.
Organization of the enchained arguments into a
proof according to current mathematical
standards 6. Approaching a formal
proof. (Boero, 1999)
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Model of the proving process - evaluation
1. Phase The first phase consists of the
exploration of the problem situation, the
generation of a conjecture and the identification
of different types of arguments for the
plausibility of this conjecture. This phase is
denoted as
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Model of the proving process - evaluation
2. Phase The second phase consists of the
precise formulation of the conjecture according
to the shared textual conventions. This phase is
denoted as
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Model of the proving process - evaluation
3. Phase This is again an explorative phase that
is based on the formulated conjecture. The aim is
the identification of appropriate arguments for
the conjecture and a rough planning of a proof
strategy. We distinguish this phase in four
subcategories (1) reference to the assumptions,
(2) investigation of the assumptions, (3)
collection of further information and (4)
generation of a proof idea. We denote this phase
as
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Model of the proving process - evaluation
4. Phase Based on the proof idea and the
selected arguments of Phase 3 it follows the
combination of these arguments into a deductive
chain that constitutes a sketch of the final
proof. This phase can be performed only verbally
or in connection with some written remarks.
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Model of the proving process - evaluation
5. Phase Here the chain of arguments of Phase 4
is written down according to the standards given
in the respective mathematics classroom. It is
important that this phase also gives a
retrospective overview about the proof process.
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Model of the proving process - results
Time-based analysis
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Model of the proving process - results
Time-based analysis
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Model of the proving process - results
Time-based analysis
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Model of the proving process - results
Quality of proof components in the lessons
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Model of the proving process - results
Quality of proof components in the classes A - D
100 treated well, 66 treated, 33 treated
badly, 0 not treated
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Model of the proving process - results
Quality of proof components in the classes E - H
100 treated well, 66 treated, 33 treated
badly, 0 not treated
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Model of the proving process - results
Analysis of Phase 3 We distinguish this phase in
four subcategories (1) reference to the
assumptions, (2) investigation of the
assumptions, (3) collection of further
information and (4) generation of a proof
idea. We investigated the proving processes for
the occurrence of theses subcategories.
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Model of the proving process - results
Analysis of Phase 3
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Model of the proving process - examples
Examples Proofs E1 and H1 Proof of the
statement Opposite sides of a parallelogram are
congruent.
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Model of the proving process - examples
Proof E1
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Model of the proving process - examples
Proof H1
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Model of the proving process - discussion

• Our analysis points up that
• essential phases in the proof process are
  neglected by the teachers.
• The teacher leads the students through the
  labyrinth of the proof situation.
• The role of the students is to guess which
  direction the teacher has in mind, so called
  fragend-entwickelnde (questioning-developing)
  teaching style.

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Model of the proving process - discussion

• There is no place for in-depth phases which are
  necessary in the proof process, e.g., for the
  exploration of the problem situation or the
  collection of additional information.
• The first exploration phase (Phase 1) consists
  mainly of making drawings and measuring the lines
  or angles.
• In the second in-depth phase (Phase 3) the
  students have no time for a deeper investigation
  or exploration of the situation.

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Model of the proving process - discussion
The consequence is that they get no real chance
to solve the proof problems on their own. They
have to follow the hints and questions of their
teacher.
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Video study Augsburg 2002 on proof instruction

• Mathematics didactical perspective
• teacher questions,
• students answers,
• interaction sequences.

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Video study Augsburg 2002 on proof instruction

• Educational perspective
• participation of students,
• teaching style,
• handling of mistakes.

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Video study Augsburg 2002 on proof instruction
Teaching style (portion of time in )
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Video study Augsburg 2002 on proof instruction
Student participation in the discourse (mean, 19
lessons)
utterances out of the video scope
13,66 impossible to categorise 7,86
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Video study Augsburg 2002 on proof instruction
Students without any participation in the
discourse ()
mean 42,9 of the students (within the scope of
the video camera)
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Video study Augsburg 2002 on proof instruction
Influence of the 20 most active students in the
discourse
mean the 20 most active students provides 52
of all contributions (within the scope of the
camera)
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Video study Augsburg 2002 on proof instruction
Example Class 3
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Video study Augsburg 2002 on proof instruction

• next steps
• analyzing interaction sequences
  (question-answer-feedback),
• describe the cognitive level of the discourse,
• combination of all video data.