Region 11 Math and Science Teacher Center
Module Overview: 9-12 Proof and Reasoning
This 9-12 Proof and Reasoning Module describes a professional development experience for
high school teachers who wish to gain a better understanding of how their students utilize
Reasoning and Proof in mathematics. The materials in the module are aligned with the
Principles and Standards for School Mathematics (NCTM) and the new Minnesota Academic
Standards for Mathematics, although it is important to note that there no specific state or
national content standards related to proof or mathematical reasoning. NCTM calls Proof and
Reasoning a process standard, reflecting the fact that mathematical proof and reasoning are
omnipresent throughout the entire mathematics curriculum.
Following the model used in Region 11’s successful 6-8 Algebra module, the module uses the
Professional Learning Community (PLC) model based on the work of Jacobs and others. This
model is driven by two key beliefs:
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For the professional development to be meaningful it needs to take place over a long
period of time.
The professional development must be explicitly related to the mathematics that the
teachers are teaching in their classrooms.
In accordance with these beliefs, this 9-12 Proof and Reasoning module is organized into five
separate sessions, each of which take place over the course of four to six weeks; completing
the entire module will therefore require the bulk of the academic year. Each session will focus
on important topics which should be relevant for all secondary mathematics teachers,
regardless of the specific courses they teach.
It can be quite difficult to choose topics which are appropriate for all high school math teachers.
At lower grade levels, most teachers instruct their students in fairly similar topics, regardless of
the particular curriculum used by their schools. At the high school level, however, one teacher
might have algebra and calculus courses, whereas another faculty member might teach only
geometry. If the sessions were organized according to courses (algebra, geometry,
precalculus, calculus) there is a real risk of certain sessions being largely irrelevant to a number
of participants. Instead, we have organized the module into five sessions which cover concepts
or types of reasoning skills which are used at all levels of high school mathematics:
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Introduction to Mathematical Proof
Introduction to Mathematical Reasoning
Algebraic and Symbolic Reasoning
Geometric and Spatial Reasoning
Data Analysis and Statistical Reasoning
Each session will begin with a large group presentation focusing on important concepts and
examples of student thinking. This presentation will be followed by four Professional Learning
Community (PLC) meetings, in which teachers will reflect on how the ideas relate to their own
classes and students. To drive their discussions, the module includes assessments and
interview protocols related to the presentation materials, allowing teachers to explore their
students’ views about mathematical proof and to diagnose their skills with various types of
reasoning.
Professional Development Module Template
Module Title: 9-12 Proof and Reasoning
Module Authors: Jonathan Rogness, Brian Lindaman
GOAL of Module: To create a professional development experience for mathematics teachers as they examine the
fundamental concepts of proof and reasoning within high school mathematics.
List of Module Outcomes:
1. Participants will learn the meaning and importance of “mathematical proof” and learn how it differs from
“proof” in other disciplines.
2. Participants will learn to identify types of mathematical reasoning and learn how to assess their students’
reasoning abilities.
3. Participants will assess student understanding to make future instructional decisions.
4. Participants will develop a deeper understanding of the mathematics they teach.
Intended Audience: Mathematics Teachers in Grades 9-12
Presentation Plan for Module:
A brief presentation plan for the module is included in the following pages. Because all of the session materials –
presentations, teacher handouts, baseline assessments, interview protocols, and summative assessments – are available
online, interested parties are encouraged to read through those materials to see the activities used during each session
and the research supporting them. An extensive bibliography is included in the “Information for Module Presenters” file.
That document also contains a more detailed description of the PLC structure.
Session 1
Time
Allocated
Sep-Oct
Mathematics
Content To
Be Delivered
Module
Overview
Teachers will
be introduced
to the NCTM
Reasoning
and Proof
standard and
have the
structure of
this module
and the PLC
activities
explained.
Mathematical
Proof
Definition of
mathematical
proof; levels of
justification /
proof used by
students;
mathematical
language used
in proofs.
Participant
Processes: How
will training on
techniques and
strategies be
provided to
teachers?
Teachers will work
through examples
designed to
differentiate
“mathematical
proof” from
“scientific proof.”
Teachers will work
through
simulated student
problems to
explore different
levels of proof,
identifying
common
misconceptions
and barriers to
student learning
along the way.
Materials To Be Used
Principles & Standards for
School Mathematics
(NCTM).
Standard exaples (e.g.
mutilated checkerboard)
used by the mathematical
community.
Articles by Knuth & Elliott
(1998) and Epp (1999)
about student
understanding of proof
and the precise language
used in proofs.
How will
PLCs and
data-driven
decision
making be
addressed?
A four meeting
work plan
related to
mathematical
proof will be
introduced to
teachers to use
in their PLCs
How will assessment
of student
performance be
addressed?
PLC work has a
specific format for
collecting formative
and summative
assessment data
related to student
understanding of
mathematical proof.
Session 2
Time
Allocated
Nov-Dec
Mathematics
Content To Be
Delivered
Mathematical
Reasoning
Different types
of mathematical
reasoning.
Participant
Processes:
How will
training on
techniques and
strategies be
provided to
teachers?
Teachers will
work through
problems which
require various
types of
reasoning,
ranging from
pattern
generalization to
logical
deduction.
Teachers will
analyze student
responses to
free-response
questions to
learn how to
assess their
students’
reasoning
abilities.
Materials To Be
Used
Peressini &
Webb’s framework
(1999) for
assessing
reasoning skills.
Analysis by Silver
& Carpenter (1989)
of reasoning ability
as tested in the
1986 Nat. Ass. of
Educ. Progress
(NAEP).
Activity based on
(Bonsague &
Gannon, 2003) to
demonstrate
necessary and
sufficient
conditions.
How will PLCs
and data-driven
decision making
be addressed?
How will assessment
of student
performance be
addressed?
A four meeting
work plan related to
mathematical
reasoning will be
introduced to
teachers to use in
their PLCs
PLC work has a specific
format for collecting
formative and
summative assessment
data related to student
understanding of
mathematical reasoning.
Session 3
Time
Allocated
Jan-Feb
Mathematics
Content To
Be Delivered
Algebraic
and Symbolic
Reasoning
Algebraic
reasoning and
errors in
algebraic
thinking.
Participant
Processes: How
will training on
techniques and
strategies be
provided to
teachers?
Materials To Be
Used
Teachers will
analyze common
types of student
errors in algebraic
reasoning.
Matz (1982)
framework for
categorizing
student errors in
algebraic thinking.
Teachers will work
through various
problems
requiring algebraic
thinking, both
symbolically and
graphically.
Student-Professor
problem described
in Schoenfeld
(1985).
Review of linear
functions activity
from NCTM OpenEnded
Mathematics text.
How will PLCs
and data-driven
decision making
be addressed?
How will assessment
of student
performance be
addressed?
A four meeting
work plan related to
algebraic and
symbolic reasoning
will be introduced to
teachers to use in
their PLCs
PLC work has a specific
format for collecting
formative and
summative assessment
data related to student
understanding of
algebraic and symbolic
reasoning.
Session 4
Time
Allocated
Mar-Apr
Mathematics
Content To Be
Delivered
Geometric
and Spatial
Reasoning
Reasoning in
geometry with
plane figures
and spatial
reasoning
about 3D
shapes.
Participant
Processes: How
will training on
techniques and
strategies be
provided to
teachers?
Teachers will
discuss the Van
Hiele levels of
geometric
understanding
and look at
student work
related to those
levels.
Some time will be
spent on the
conditions for a
visual proof, and
various proofs of
the Pythagorean
Theorem will be
explored and
compared.
Teachers will
explore isometric
perspectives of
3-d objects.
Materials To Be
Used
Van Hiele levels of
geometric
understanding.
Conditions for
visual proofs
(Hanna & Sidoli,
2007).
Selected material
from book Proofs
Without Words,
MAA, 1993.
Adapted isometric
activity from
Navigating through
Geometry in
Grades 6-8,
NCTM, 2002.
How will PLCs
and data-driven
decision making
be addressed?
How will assessment
of student
performance be
addressed?
A four meeting
work plan related to
geometric and
spatial reasoning
will be introduced to
teachers to use in
their PLCs
PLC work has a specific
format for collecting
formative and
summative assessment
data related to student
understanding of
geometric and spatial
reasoning.
Session 5
Time
Allocated
May-Jun
Mathematics
Content To Be
Delivered
Data Analysis
and Statistical
Reasoning
Collecting data
and performing
a statistical
analysis
involves skills
which are
related to, but
quite different
from the
mathematical
reasoning
discussed in
previous
sessions.
Participant
Processes: How
will training on
techniques and
strategies be
provided to
teachers?
Teachers will
take the
Statistical
Reasoning
Assessment
(SRA) and
analyze their
responses to
diagnose their
own abilities in
preparation for
working with their
students.
Teachers will
also work
through a data
analysis
simulation.
Materials To Be
Used
Statistical
Reasoning
Assessment
developed by Joan
Garfield (2003).
Navigating through
Reasoning and
Proof in Grades 912 (NCTM, 2008).
How will PLCs
and data-driven
decision making
be addressed?
How will assessment
of student
performance be
addressed?
A four meeting
work plan related to
data analysis and
statistical reasoning
will be introduced to
teachers to use in
their PLCs
PLC work has a specific
format for collecting
formative and
summative assessment
data related to student
understanding of data
analysis and statistical
reasoning.