Tasks

Students are given a conditional statement and must decide whether three "proofs" do or do not prove the statement.

An introductory task that supports students' understanding of what counts as a mathematically valid argument.

A task for introducing students to proving existence statements nonconstructively.

A task for promoting students' distinction between the role of the variable k and the variable n in proof by Mathematical Induction.

A geometric task engaging students in unpacking how the order of quantification impacts the mathematical logic of a given statement.

An introductory task prompting students to think about how to show a conditional statement is false. This is a first step toward developing the precise negation of an implication.

A task for transitioning students from understanding how order of quantification affects mathematical logic to formally proving multiply quantified statements.

This task gives students a first look at multiply quantified statements through five example statements.

This task provides students with a first look at the existential quantifier. Students must determine when an existential statement is true and how they might prove it.

This task introduces students to the universal quantifier. Students must determine when a universally quantified statement is true and how they might prove it.

This task supports students' understanding of why a logical implication and its contrapositive are logically equivalent.

This introductory task on negation is intended to be given before stating the formal definition of negation.

This task is intended to address students issues with quantifying the negation. It should be given after both an introductory task on and the formal definition of negation.

This task is designed to support students construction of functions as objects. Given properties of the composition of two functions, they must decide whether the individual functions must inherit these same properties.

This is an initial task for motivating students to recognize that a constructive existence proof should simply introduce the existent value and demonstrate that it works. This notion often runs orthogonal to students' past experience of showing the solution process for solving for an unknown in an algebraic equation.

This is an introductory task that can be given to students on the first day of class to begin establishing the sociomathematical norm of translating mathematical symbols and language into meaningful understanding.

This task is a first introduction to (open) mathematical statements. Students are given an open expression and an open statement and must discuss whether they are mathematical statements. It encourages students to notice the importance of quantifying variables.

In this tasks, students are given pairs of open statements and asked to determine the relationship between their truth sets via generating their own Euler diagrams. This can be used as a first task after students have been given the definition of the truth set of an open statement and been introduced to visualizing truth sets via Euler diagrams.

This introductory task should be used immediately following statement of the definition of a logical implication (or conditional statement). Students are given quartets of conditional statements to compare and contrast. This supports students to develop an understanding of logical implication based on truth sets represented through Euler diagrams. Students are invited to notice and abstract common logical structure across all three quartets.

This task engages students with the truth of an implication with a false hypothesis.

This task supports students' understanding of a vacuously true implication through the use of an Euler diagram. Pairs well as a follow-up to "Using Euler Diagrams to Understand a Vacuously True Implication".

This task supports students' understanding of the relationship between the truth set of an implication and its negation.

These tasks comprise an entire instructional unit on the principle of mathematical induction, where students reinvent the necessary logical components of proof by induction and reason about the necessity of the base case, the logic of the inductive implication, as well as determining the appropriate quantification.