Proof Comprehension Tests

We have created two sets of proof comprehension tests that can be used for instructional and research purposes. There are three tests designed for the Introduction to Proof course and eight tests for the Real Analysis level. 

Proof Comprehension Assessment Framework

Mejia-Ramos et al (2013) developed a framework for proof comrehension assessment, which is outlined in the table below. We used this framework to develop short, multiple-choice tests for a variety of proofs in an Introduction to Proof course and an undergraduate Real Analysis course. If you are interested in using these tests for instructional or research purposes, you can request them at the link at the bottom of this webpage. 

Facet

Description

Local

 

Meaning of terms and statements

Items of this type measure students’ understanding of key terms and statements in the proof.

Logical status of statements and proof framework

Items of this type assess students’ knowledge of the logical status of statements in the proof and the logical relationship between these statements and the statement being proven.

Justification of claims

Items of this type address students’ comprehension of how each assertion in the proof follows from previous statements in the proof and other proven or assumed statements.

Holistic

 

Summarising via high-level ideas

Items of this type measure students’ grasp of the main idea of the proof and its overarching approach.

Identifying the modular structure

Items of this type address students’ comprehension of the proof in terms of its main components/modules and the logical relationship between them.

Transferring the general ideas or methods to another context

Items of this type assess students’ ability to adapt the ideas and procedures of the proof to solve other proving tasks.

Illustrating with examples

Items of this type measure students’ understanding of the proof in terms of its relationship to specific examples.

Introduction to Proof Comprehension Tests

First page of one of the ITP proof comprehension tests

For the Introduction to Proof course (or similar courses), we have developed a set of three proof comprehension tests. The topics of the tests are:

  1. Infinitude of Primes
  2. Fibonacci parity
  3. Uncountability of the open interval (0,1).

For more on the development of the Introduction to Proof tests, see Mejia-Ramos et al (2017). 

Real Analysis Comprehension Tests

For the undergraduate Real Analysis course (or similar courses), we have developed a set of eight proof comprehension tests. The topics of the tests are:

  1. Nested Interval Property
  2. Monotone Convergence
  3. Bolzano-Weierstrass Theorem
  4. Cauchy Criterion
  5. Sequential Convergence
  6. Extreme Value Theorem
  7. Intermediate Value Theorem
  8. Mean Value Theorem

 

Proof from one of the Real Analysis Proof Comprehension Tests

Project Teams

Introduction to Proof Comprehension Tests

Juan Pablo Mejia Ramos, Rutgers University

Keith Weber, Rutgers University

Kristen Lew, Texas State University

Jimmy de la Torre, The University of Hong Kong

Real Analysis Comprehension Tests

Juan Pablo Mejía Ramos, Rutgers University

Kate Melhuish, Texas State University

Kristen Lew, Texas State University

Keith Weber, Rutgers University

Drew Gitomer, Rutgers University

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