Mathematical Reasoning

Mathematical Proof: Review & Sharing

What is Mathematical Reasoning?

Inductive and Deductive Reasoning

How to Assess Reasoning Skills

Problem Solving: Exploring to Generalization

Today’s Agenda

Proof ReviewProof Review

It is “proofs rather than theorems that are the bearers of mathematical knowledge”

(Rav, 1999)

Proof ReviewProof ReviewPass around and look at student baseline

assessments within your group.

Assess and sort according to students’ understanding of proof and language◦ 0: no understanding◦ 1: some understanding (use of examples)◦ 2: understands the need for proof and attempts

abstract arguments, but struggles◦ 3: high level understanding

Describe some of your students’ views on justification and proof

Proof ReviewProof Review

What did you learn most from the Proof baseline and summative assessments?

Describe at least two ways that you helped your students understand what mathematical proof is.

Describe one classroom situation where you saw a student exhibit growth in thinking about proof.

What is What is Mathematical Mathematical Reasoning?Reasoning?

Initial DiscussionInitial DiscussionWith your neighbors at your table, try to write a definition of of mathematical reasoning.

How does it differ from mathematical proof?

DefinitionDefinition

Mathematical reasoning involves gathering evidence, building arguments, and drawing logical conclusions about these various ideas and their relationships.

Types of Reasoning

There are many types of mathematical reasoning, including:

•Inductive (e.g. pattern generalization)

•Deductive (logic)

•Geometric

•Algebraic

Can your group think of any others?

Inductive and Inductive and Deductive Deductive ReasoningReasoning

PISA Apple Orchard Problem

Logic PuzzleJerell has three children, Jordan, Nastasia and Logan. They attend Central, Humboldt and Arlington and play Soccer, Tennis and Basketball.

a. Jordan does not attend Central.b. Nastasia is not at Humboldt.c. The child at Central does not play soccer.d. The student at Humboldt plays tennis.e. Nastasia does not play basketball.

Which child attends which school, and what sport does each one play?

Mathematical Methods

Some versions of the NAEP assessment have marked certain problems as measuring “mathematical methods.”

( = “mathematical reasoning” )

(Silver & Carpenter, 1989) analyzed the results from the 1986 NAEP exam.

Based on this limited data, what conjectures might you make about students’ reasoning abilities in different contexts and with different types of problems?

NAEP Discussion

How to Assess How to Assess Reasoning SkillsReasoning Skills

Analyzing Reasoning SkillsAnalyzing Reasoning Skills

ADW: Analysis of Paul

ADW: Analysis of Ben

RND: Analysis of Paul

RND: Analysis of Ben

SAS: Analysis of Paul

SAS: Analysis of Ben

Synthesis

Work with your table to create a summary of Paul’s and Ben’s abilities in the following areas:

•Foundational Knowledge•Mathematical Reasoning•Communication

Then create a condensed summary of their overall reasoning abilities with help from your colleagues and the presenter.

Problem Solving: Problem Solving: Exploring to Exploring to

GeneralizationGeneralization

Problem

Which natural numbers can be written as the sum of two or more consecutive natural numbers?

Experiment on the table in your handout!

BaselineBaselineAssessmentAssessment

Baseline Assessment

Baseline Assessment

Special AssignmentSpecial Assignment

The baseline assessment contains one of the problems you analyzed as part of the Peressini & Webb framework. The Interview Protocol includes the other two.

Choose a variety of students for your interviews. Combined with their baseline assessments, you have all three of the problems we used to analyze Paul and Ben’s reasoning abilities. Perform the same analysis for the students you interview, using all three problems.