{"id":8406,"date":"2025-08-26T15:58:02","date_gmt":"2025-08-26T19:58:02","guid":{"rendered":"https:\/\/www.bu.edu\/teaching-writing\/?page_id=8406"},"modified":"2025-09-03T15:40:15","modified_gmt":"2025-09-03T19:40:15","slug":"beginning-proof-writing","status":"publish","type":"page","link":"https:\/\/www.bu.edu\/teaching-writing\/teaching-writing-at-bu\/writing-in-the-disciplines\/teaching-writing-in-math-and-statistics\/math-resources\/audience-genre-in-mathematical-writing\/beginning-proof-writing\/","title":{"rendered":"Beginning Activities for Proof Writing"},"content":{"rendered":"

Objective: <\/b>Students will learn foundational skills related to proof writing.\u00a0<\/span><\/p>\n

Key Terms: <\/b>Proof, induction, negation, proof by way of contradiction.<\/span><\/p>\n

Timing: <\/b>\u00a0These lessons could be appropriate when students are first learning to write proofs, or for students who have had some experience with proof writing but are still struggling with foundational aspects of it.\u00a0<\/span><\/p>\n

Conceptual Framework<\/h6>
When students first start writing proofs, it can be challenging for them to know where to start. This is particularly true if they are used to more formulaic or algorithmic problem solving, and less used to more open-ended problems where there is more space for creativity. These lessons focus on the foundational aspects of proof writing and discuss common proof structures (eg induction, proof by way of contradiction) in settings where the math itself is not too involved, so that the focus can be on learning to write proofs, rather than on learning new content.\u00a0<\/span><\/p>\n

<\/div>\n<\/div>\n<\/p>\n

Lesson<\/b><\/h2>\n

Part I: Fill-in-the-blank proofs<\/span>\u00a0<\/span><\/p>\n

    \n
  1. Introduce<\/strong> the students to a particular proof technique, such as proof by induction, and include at least one example of such a proof.<\/span><\/li>\n
  2. Determine<\/strong> a new statement that you would like the students to prove using that technique. Write your own proof of that statement, but remove some of the key details and replace them with blanks that the students must fill in. See, for example, this <\/span>example from a discrete math course<\/span><\/a>, written by Debbie Borkovitz.\u00a0<\/span><\/li>\n
  3. This activity could be done during class, or outside of class. Student chould first try to complete the activity on their own, and that individual work could be followed with a discussion<\/strong> in groups where students each share how they completed the proof. If you utilize this group discussion, you could have the groups share out what they discussed, and you could use this opportunity to point out not that there is not necessarily just one correct way to fill in the blanks.<\/span><\/li>\n
  4. This activity could be followed with additional exercises<\/strong> where less and less of the text is pre-populated for them, and they need to fill in more and more of the proofs on their own.\u00a0<\/span><\/li>\n<\/ol>\n

    Part II: Setting up proofs by contradiction.\u00a0<\/span><\/p>\n

      \n
    1. Consider<\/strong> this <\/span>worksheet on negations<\/span><\/a>, created by Margaret Beck. You may wish to modify the mathematical examples in the second part of the worksheet to better match the content of your course.\u00a0<\/span><\/li>\n
    2. Put students in small groups<\/strong>. Ask them to spend a few minutes trying to negate each statement on their own. Then give them additional time to discuss their negations within their group.<\/span><\/li>\n
    3. Ask the groups to share<\/strong> their negations for each statement. You could, for example, assign each group one or two of the negations that they will need to present to the class, such as by writing their negation on the board.<\/span><\/li>\n
    4. Expand upon and\/or synthesize<\/strong> student comments, eg. at the board. Explain that, in order to write a successful proof by way of contradiction, one must first understand how to negate the assertion that is being made.\u00a0<\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

      Objective: Students will learn foundational skills related to proof writing.\u00a0 Key Terms: Proof, induction, negation, proof by way of contradiction. Timing: \u00a0These lessons could be appropriate when students are first learning to write proofs, or for students who have had some experience with proof writing but are still struggling with foundational aspects of it.\u00a0 Lesson […]<\/p>\n","protected":false},"author":15811,"featured_media":0,"parent":8461,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"https:\/\/www.bu.edu\/teaching-writing\/wp-json\/wp\/v2\/pages\/8406"}],"collection":[{"href":"https:\/\/www.bu.edu\/teaching-writing\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.bu.edu\/teaching-writing\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.bu.edu\/teaching-writing\/wp-json\/wp\/v2\/users\/15811"}],"replies":[{"embeddable":true,"href":"https:\/\/www.bu.edu\/teaching-writing\/wp-json\/wp\/v2\/comments?post=8406"}],"version-history":[{"count":5,"href":"https:\/\/www.bu.edu\/teaching-writing\/wp-json\/wp\/v2\/pages\/8406\/revisions"}],"predecessor-version":[{"id":8730,"href":"https:\/\/www.bu.edu\/teaching-writing\/wp-json\/wp\/v2\/pages\/8406\/revisions\/8730"}],"up":[{"embeddable":true,"href":"https:\/\/www.bu.edu\/teaching-writing\/wp-json\/wp\/v2\/pages\/8461"}],"wp:attachment":[{"href":"https:\/\/www.bu.edu\/teaching-writing\/wp-json\/wp\/v2\/media?parent=8406"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}