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Proof for Everyone

Using a Journal Article as a Professional Development Experience
Reasoning and Proof 

PDF 

Title:           Proof for Everyone
Author:        Eugene A. Olmstead
Journal:       Mathematics Teacher
Issue:          February 2007, Volume 100, Issue 6, pp. 436-439
     
Title:            The Surfer Problem: A “Whys” Approach
Authors:       Larry Copes and Jeremy Kahan
Journal:       Mathematics Teacher
Issue:          August 2006, Volume 100, Issue 1, pp.14-19

 

Rationale/Suggestions for Use 

Reasoning and Proof are often confined to high school geometry classes and proof is often viewed narrowly as a two-column proof. The Reasoning and Proof Standard (2000) states, “Reasoning and proof are not special activities reserved for special times or special topics in the curriculum but should be a natural, ongoing part of classroom discussions, no matter what topic is being studied.” (p. 342). The two articles were chosen to emphasize the role and function of proof – one article suggests how this might look in a geometry class while the other article considers the role of reasoning and proof in algebra. Both articles describe the process of investigating and conjecturing, exploring the problem inductively before doing a deductive proof. The content is appropriate for a wide variety of educators at the secondary level.

Procedures/ Discussion Questions 

Goal: Participants will compare the role of reasoning and proof in algebra and geometry.

  1. NOTE: Prepare ahead of time the two problems on a separate handout so participants can work on their problem without reading the article first. Other materials and supplies may be needed (see Algebra Group part “b” and Geometry Group part “a”). You will also need a page with the quotes from the Reasoning and Proof Standard (see #5 below) for the final session.
  2. Either by random assignment or participant preference, divide the group into two smaller groups. One group will work with the article on reasoning and proof in geometry (“The Surfer Problem…”) while the second group will work with the article on reasoning and proof in algebra (“Proof for Everyone” - the parabola problem).
  3. Algebra Group:  
    • Present the parabola (Proof_1) problem to participants. This can be accomplished by copying just page 437 from the article without the “Stage 2” section (the advantage here is that all the directions for using a graphing calculator or dynamic software as well as directions for graphing on a grid are included), or summarize the problem as follows: “Graph Proof_1 carefully plotting the integer coordinates of the x-values from -10 to 10. Draw a segment connecting any two of the integer values for the endpoints, one from the left side of the parabola and one from the right side of the parabola. Consider several cases and identify any patterns that you notice. Organize your work so others can follow your thinking.” Use the additional prompts, “Where does the segment cross the y-axis?” and “What is the slope of the segment?” as needed. * Ask participants to work individually on the problem. They will need a graphing calculator, graphing software, and/or grid paper.  Emphasize the directions to select integer values in order to see patterns more easily and if they are using graph paper, to be as precise as possible including using a straight edge to draw the segment. Given the nature of the problem, some kind of exploration tool/software might be helpful but not necessary.
    • After a few minutes of private think time, ask participants to share their thinking with a partner or in table groups. (Use a partner share or a go-around protocol to be sure all ideas are heard before the group starts to discuss the problem.)
    • Allow groups to continue working on the problem to come up with a conjecture before they move on. (You may need to set a time limit, however.) Ask them to try their conjecture on several specific cases and record their thinking in a way others can follow it.
    • Next, ask the groups to work collaboratively to prove the conjecture for all real numbers. The facilitator may need to provide some scaffolding, but you don’t want participants to read the article with sample proofs before they have made the attempt to prove the case themselves. If needed, suggest they write out their conjecture using Proof_2 and proof_3 as two points on the curve.
    • If time allows, or if one group finishes before the others, suggest they work on the proof of the general case proof_4. Note, if there is not time for groups to pursue this case, it is not necessary for the discussion.
    • If you have more than one “algebra” group, provide time for the two groups to share conjectures and proofs. This could be accomplished by pairing one person in each group with a person from the other group.
    • Participants should conclude by reading the article and consider the following questions as they relate to their experience:
      • In what ways does the author motivate the students to be curious and ask questions/be interested in proving their conjectures?
      • How was this experience similar to or different from “reasoning and proof” as you experienced it in your high school work?
      • In what ways does the article connect to and extend your thinking about The Reasoning and Proof Standard which says: “Instructional programs from prekindergarten through grade 12 should enable all students to –
        • Recognize reasoning and proof as fundamental aspects of mathematics;
        • Make and investigate mathematical conjectures;* Develop and evaluate mathematical arguments and proofs;
        • Select and use various types of reasoning and methods of proof.”
  4. Geometry Group: 
    • Present the “surfer problem” to the participants (copy from page 14 of the article and/or put it on chart paper). Provide materials for participants to physically model the problem as noted in the article (string, tape measures, protractors, and sticky notes). As in the article, suggest they do not use the Geometer’s Sketchpad or other exploration programs at this time, but rather that they work together to model the problem and keep track of their thinking by recording measurements and drawings in a journal and that they make conjectures based on their model(s).* Through the physical model, participants should conjecture that it doesn’t matter where the hut is located. Ask them to consider, “Why doesn’t it matter?” and “Can you prove that it doesn’t matter where the hut is located?” (Note: if there are other conjectures, suggest they put those on ‘hold’ until they have had a chance to investigate this one as it is the one the article focuses on.)* If you have more than one “geometry” group, provide time for the two groups to share conjectures and proofs. This could be accomplished by pairing one person in each group with a person from the other group.* Participants should conclude by reading the article, studying the proofs provided in the article, and considering the following questions as they relate to their experience:
      • In what ways does the author motivate the students to be curious and ask questions/be interested in proving their conjectures?
      • How was this experience similar to or different from “reasoning and proof” as you experienced it in your high school work?
      • In what ways does the article connect to and extend your thinking about The Reasoning and Proof Standard which says: “Instructional programs from prekindergarten through grade 12 should enable all students to –
        • Recognize reasoning and proof as fundamental aspects of mathematics;
        • Make and investigate mathematical conjectures;
        • Develop and evaluate mathematical arguments and proofs;
        • Select and use various types of reasoning and methods of proof.”
  5. Bring the two groups back together. Before facilitating a whole group discussion, ask participants to pair up with a person from the other group (so you have pairs  - one from the algebra group and one from the geometry group). Distribute a copy of the quotes from the Reasoning and Proof Standard below. Ask each pair to discuss the quotes and select two that best represent their understanding of the Standard and how it is portrayed in the articles. Be prepared to share specific references from the articles or your experience with the mathematics in the articles that led you to select that quote.

    Quotes from the Reasoning and Proof Standard  
    • “The habit of asking why is essential for students to develop sound mathematical reasoning.” (p. 344)
    • “Mathematics should make sense to students; they should see it as reasoned and reasonable. Their experience in school should help them recognize that seeking and finding explanations for the patterns they observe and the procedures they use help them develop deeper understandings of mathematics.” (p. 342)
    • “Sometimes developing a proof is a natural way of thinking through a problem.” (p. 344)
    • “The repertoire of proof techniques that students understand and use should expand through the high school year.” (p. 345)
    • “(Students) should be able to produce logical arguments and present formal proofs that effectively explain their reasoning, whether in paragraph, two-column, or some other form of proof.” (p. 345)
    • “Students should reason in a wide range of mathematical and applied settings.” (p. 345)
    • “Through the classroom environments they create, mathematics teachers should convey the importance of knowing the reasons for mathematical patterns and truths.” (p. 345)
    • “… students must develop enough confidence in their reasoning abilities to question others’ mathematical arguments as well as their own. In this way, they rely more on logic than on external authority to determine the soundness of a mathematical argument.” (p. 346)
  6. Facilitate a whole group discussion by asking each pair to share their favorite quote (or second favorite if their first has already been shared) and why they selected it. Facilitator should chart these ideas.
  7. Allow a few minutes private think time for folks to respond to the question, “What are the implications for our practice?”, and then ask participants to share their ideas first with a partner and then with the whole group. Facilitator should chart these for future reference. Press on the idea of reasoning and proof in other content areas besides geometry.

Next Steps 

  • Participants use the problems with at least one class and bring back student work to analyze at the next session.
  • Long term, the group might consider analyzing their current courses to determine where students are encouraged to ask the question, “Why does it work?” and where they might enhance the curriculum to provide more opportunities for students to investigate a mathematical idea, conjecture and then develop proof strategies.

Connections to Other NCTM Publications 

 

 

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