Writing precisely with quantifiers: which details matter

Objective: Within the context of quantifiers, students will be able to identify the details in their writing that affect the mathematical meaning of a sentence, and the details in their writing that are less consequential in determining the overall success of the proof.  

Key Terms: Proof, notation, variable, correctness, quantifier.

Timing: These lessons could be appropriate in classes where any students are gaining familiarity with proof writing, even if that only applies to a minority of students. Consider trying these exercises early in the semester but after you have introduced some content.

Lesson

Part I: Quantifiers

1. Consider this worksheet, Quantifiers in the Context of Sequential Convergence, created by Margaret Beck. You may wish to modify the example it contains to better match the content of your course. Also, note that the solutions are on the second page, so don’t distribute those to the students (yet).
2. Put students in small groups. Ask them to spend a few minutes completing the activity on their own. Then give them additional time to discuss their work within their group.
3. Ask the groups to share what they discussed.
4. Expand upon and/or synthesize student comments, eg. at the board. Possibly also share and/or distribute solutions. 

Part II: Nested Quantifiers

1. Consider this worksheet on Nested Quantifiers, created by Margaret Beck. You may wish to modify the example it contains to better match the content of your course. Also, note that the solutions begin on the second page, so don’t distribute those to the students (yet).
2. Put students in small groups. Ask them to spend a few minutes completing the activity on their own. Then give them additional time to discuss their work within their group.
3. Ask the groups to share what they discussed.
4. Expand upon and/or synthesize student comments, eg. at the board. Possibly also share and/or distribute solutions.