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:

  • 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:

  • 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 / Mathematics Content To Be Delivered / Participant Processes: How will training on techniques and strategies be provided to teachers? / Materials To Be Used / How will PLCs and data-driven decision making be addressed? / How will assessment of student performance be addressed?
Sep-Oct / 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. / 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. / 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. / A four meeting work plan related to mathematical proof 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 proof.

Session 2

Time
Allocated / Mathematics Content To Be Delivered / Participant Processes: How will training on techniques and strategies be provided to teachers? / Materials To Be Used / How will PLCs and data-driven decision making be addressed? / How will assessment of student performance be addressed?
Nov-Dec / Mathematical Reasoning
Different types of mathematical reasoning. / 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. / 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. / 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 / Mathematics Content To Be Delivered / Participant Processes: How will training on techniques and strategies be provided to teachers? / Materials To Be Used / How will PLCs and data-driven decision making be addressed? / How will assessment of student performance be addressed?
Jan-Feb / Algebraic and Symbolic Reasoning
Algebraic reasoning and errors in algebraic thinking. / Teachers will analyze common types of student errors in algebraic reasoning.
Teachers will work through various problems requiring algebraic thinking, both symbolically and graphically. / Matz (1982) framework for categorizing student errors in algebraic thinking.
Student-Professor problem described in Schoenfeld (1985).
Review of linear functions activity from NCTM Open-Ended Mathematics text. / 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 / Mathematics Content To Be Delivered / Participant Processes: How will training on techniques and strategies be provided to teachers? / Materials To Be Used / How will PLCs and data-driven decision making be addressed? / How will assessment of student performance be addressed?
Mar-Apr / Geometric and Spatial Reasoning
Reasoning in geometry with plane figures and spatial reasoning about 3D shapes. / 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. / 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. / 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 / Mathematics Content To Be Delivered / Participant Processes: How will training on techniques and strategies be provided to teachers? / Materials To Be Used / How will PLCs and data-driven decision making be addressed? / How will assessment of student performance be addressed?
May-Jun / 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. / 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. / Statistical Reasoning Assessment developed by Joan Garfield (2003).
Navigating through Reasoning and Proof in Grades 9-12 (NCTM, 2008). / 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.