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Moore method?

Chp 20 Formal-rhetorical: knowing proof schemes; having a hierarchical structure. Problem-centered reasoning: "may call on conceptual knowledge, mathematical intuition, and the ability to bring to mind the "right" resources at the "right time" (p 344)

Chp 21 ehhhh

An Analytic Framework of reasoning and proving Picks up on Chapter 14 work. Is there a connection between Selden's work and the framework?

Chp 16 DNR: do not resuscitate! p 280: "Proofs really aren't there to convince you that something is true - they're there to show you why it is true." (Yandell, 2002, p 150).

Chp 17 Proof as a social activity or process.

Chapter 14 We liked the framework posed by Silver & Stylianides. We understand the difference between plausible and definite. Why do students have difficulty in trusting patterns? Well, sometimes they should when the pattern is merely plausible.

Chapter 9 Christine was bothered with the validity of many of the constructs. AJ didn't like the tasks; were they even piloted?? Prompts were inconsistent. On p 170, Laura noticed the importance of the curriculum support materials that can help teachers take advantage of the classroom opportunities for students' understanding justification.

For June10, we are reading chapters 9 and 14. We can plan on meeting June 17 and 24. • On June 24, in addition to any final readings we will discuss, Jon & I would like all (but particularly the graduate students) to share what they have gleaned from the readings in the Book Club and if they have made any connections of these readings to their current research interests, along with any other life-changing experiences these readings have prompted. : ) • Finally, for those who have indicated an interest in writing up a book review, we will meet to discuss plans for that task at the end of or immediately following the June 24 meeting. UPDATE: Jon just found out that Hy Bass beat us to the book review so we won't be doing this one!

Is proof more than sense-making? Post a reply in the discussion.

logic; reasoning, structured reasoning, empirical reasoning, a line of reasoning; argument, conclusive argument, argument for defending, for disagreement, for justification, for refinement; mathematical argument, different, sufficient, sophisticated, acceptable mathematical argument; justification; explanation; proof, rigorous proof, informal proof, formal proof, deductive proof, inductive proof; phases of proof: convince yourself, convince a friend, convince an enemy (convince a skeptic); proof is relative to the context; mathematical legality, mathematical strategy; data to conclusion with warrant and backing; hypotheses; conjectures; premises; refutations; counterexamples; mathematical knowing; sense-making
 * Terms/ideas:**


 * CHAPTER SUMMARIES**


 * Deborah Schifter, Representation-based Proof in the Elementary Grades

Young children are capable of justifying claims of generality and visual representations can assist students in making these claims. The author found that these representations needed to meet three criteria:

1. The meaning of the operation(s) involved is represented in diagrams, manipulatives or story contexts. 2. The representation can accommodate a class of instances (for example, all whole numbers). 3. The conclusion of the claim follows from the structure of the representation. **

The central argument of this chapter is that schematics for mathematical relationships that are tied to familiar actions can bolster children's ability to prove (or engage in deductive argument). To exemplify this idea, Morris discusses features of an elementary school curriculum from Russia that incorporates such schematics into its design. Examples of part-whole relationships, multiplicative relationships, and creating expressions from missing quantities are given. The intent of these schematics is to help students understand relationships between classes of objects so that children are more successful at engaging in deductive reasoning that draws on general relationships, as opposed to many specific examples. --NF
 * Anne K. Morris, Representations that Enable Children to Engage in Deductive Argument, pp. 87-101 (Chapter 5)**

One specific example from Chapter 5 that helps to expand one's understanding of this chapter is when 1st grade students work on a task of pouring water from a bucket, named A, into two other buckets, named B and C. Students do not know any quantity measure of water that is poured, just know that a single container was poured into two other containers. The “whole” bucket A and the two “part” buckets B and C are represented on a tree diagram/lattice to depict this generalized action. Students using this schematic reason by reading up and down, depending on how they want to relate the whole/part containers. Operations follow from this schematic by students writing relationships using algebraic statements. After this process, students apply their understanding to specific quantities to determine the “whole” or “part” quantities instead of A, B, and C. (AJ, 5/27/09)


 * Carolyn A. Maher, Children's Reasoning: Discovering the Idea of Mathematical Proof, pp. 120 - 132 (Chapter 7)**

Maher shares her work with young children (beginning with 8 yr olds) on how they can develop convincing arguments naturally when placed in learning environments that support such an activity. "Proof making is a special type of mathematical activity in which children attempt to justify their claims by deductive argumentation. In our view, students' learning of proof making occurs as they develop more sophisticated representations of mathematical argumentation" (p 121). The chapter provides the analysis of videotape transcripts from two 90 minute sessions of children working in pairs and as a class as they developed justifications for their solutions to problems centered around building a collection of block towers.

The students in Maher's study were not necessarily learning proof, so much as they were learning about proof. They were working with unifix cubes as a context for their reasoning. They were making block towers, each tower containing four blocks, using red and yellow blocks. The question they had to answer was, how many different possible towers are there using these two colors? It seemed to be a good problem for students to work on because there were multiple ways to find the answer and there are enough possibilities that the task is not trivial (for a third grader). In fact, one pair of girls came up with the correct //number// of towers, but was missing one tower and had another tower repeated. The students were not told they were wrong, but instead were asked by the researcher how they knew they had all of the towers. The girls then came up with some interesting ways to organize the towers, found their error, and provided arguments (one recursive, the other based on examining patterns within and across groups of towers) for why they had now found all the towers.

Maher provides examples of students who offer both recursive arguments and proofs by contradiction in the paper. I had some difficulty seeing the argument labeled as proof by contradiction as an actual proof by contradiction, even after considering the mathematical community at work (third graders). I don’t see this as a shortcoming of what Maher did in the classroom, but I wonder if it is always appropriate to look for trappings of advanced mathematics in elementary classrooms. Maybe there is a need for descriptions of mathematical arguments at the K-5 level?

Regardless, Maher does not claim that the students knew these were the types of arguments they were making and in her conclusion emphasizes that the children learned about the idea of proof. She suggests it may be more effective if teachers focus on children's natural interest in sense-making and promote that in the classroom rather than focusing on having all children providing proofs, especially if they have no idea what proving means. (Dr. B. & Rob, 5/27/09)


 * David A. Reid and Vicki Zack, Aspects of Teaching Proving in Upper Elementary School, pp 133-146 (Chapter 8)**

This chapter focused on reasoning and proving in a grade 5 classroom, where the tasks themselves came from POWs. Four POWs were the focus of the chapter: Count the Squares, Prairie Dog Tunnels, Handshakes, and Decagon Diagonals. A bulk of the analysis from this chapter concentrated on the first problem (How many squares are in a 4x4 square figure? A 5x5 square figures? 10x10? 60x60?). In analyzing how the teacher intervened during student conjecturing, one author (the teacher in this case) requested clarification of specific points (clarity), asked for explanations (explanation), and reminded students to allow others to fully participate (attention to others were valued). The teacher also ensured her classroom had three characteristics imperative to the teaching of proving: conjecturing, leaving the criteria for a correct solution up to the students, and expectations for communication. (JE, 5/27/09)

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Ideas to ponder:

1. A student once asked a mathematician at the University of Wisconsin - Madison who was also an editor for a prestigious mathematics journal for a definition of a proof. He replied, "A proof is whatever I say it is." Interesting. I'm wondering if we could edit the above to say "A student once asked a mathematician at (fill in the blank) for a definition of proof." The rest can remain unchanged!! Is this similar to the following logic? "Beauty is in the eye of the beholder". I have read that the Pythagorean view is beauty consists in mathematical perfection.* Therefore, mathematical perfection is in the eye of the beholder? or "A proof is whatever I say it is"? Okay, perhaps stretching it somewhat......

So, I can't help but wonder why his position as an editor is important. Is he asserting his status in an arrogant way? Or is he simply stating what we seem to agree on (?), that proof is an argument that convinces your mathematical community. Since he is the editor of a prestigious mathematics journal, perhaps he is a worthy representative of the mathematical community served by that journal? Anyway, it doesn't seem like a very helpful thing to say ... I imagine that most mathematicians would try to be a bit more thoughtful. (Rob, 5/27/09)


 * If you want a reference, ask Me.

2. In //Proofs and Refutations// (Lakatos, 1976) a student responds, "Then 'the virtue of a logical proof is not that it compels belief, but that it suggests doubts'" (p. 48).

3. Do you agree that Lampert presented an " 'existence proof' that certain kinds of knowing and learning are possible in the school setting under //ordinary// conditions…" (italics added)? Were these ordinary conditions?

=NEWS ITEM:= Jon and I are wondering if any of you in the group would like to work together to compose a review of this book for JRME?? Let us know and we'll put together some plan!

Jonathan, Nicole, and AJ are interested. What is required of a JRME book review? When would the writing be taking place? Here is a website with info on the criteria, etc, for a JRME book review: http://www.nctm.org/publications/content.aspx?id=17169

=CLICK THE MESSAGES PAGE FOR 5/22=

- -- Comments: I was thinking on where to put logic into our talk of proof. I just read somewhere else "the ground rules of reasoning (which is a part of proving) is **logic**". So if we don't use logic, we can't construct a proof? Are there levels of understanding and use of logic?

Here are very rough notes from May 6th. Please edit Series Editor’s Foreword Alan Schoenfeld •problem with picking points on a circle (p xiii) and seeing that after 5 cases, the pattern no longer holds that one thought might be a proof. •visual proof: the extent to which the language surrounds the picture explanatory power of proof; you could explain proof. Good proof teaches as well as convinces. Why prove it if I’m already convinced? What is the role of proof? •phases of proof: convince yourself, convince a friend, convince an enemy (convince a skeptic) •Cecilia Hoyles work in GB: looking at levels of the audience in assessment; developing the craft of argumentation •proof/argumentation; a line of reasoning is proof • reasoning is a component of proof

Chapter 2 Chazan and Lueke Our model suggests to us that justifying the legality of steps in a method is less likely to occur in a situation in which teachers and students have the roles and responsibilities we have laid out.

mathematical legality and mathematical strategy

What are the actions going on in proof? Reasoning has to take place. Memorize vs proof. Students need to muck around with it. Proof is an argument that a conclusion follows logically from an accepted given. Establish the validity of a statement. Has to be personal investment.

What is a proof and what does it look like when someone is proving?

p 31 a method is not an algorithm because there are some small level decisions to make about how to carry it out. p 35 our argument is that the centrality of method in the situation of solving of equations creates opportunities for reasoning about the strategic utility of particular steps, but suppresses opportunities for justification that particular steps are indeed mathematically legal.

The authors claim to examine one design of instruction meant to change classroom norms to match disciplinary norms more closely. Make room for mathematical reasoning in the classroom. Providing a method vs a specific strategy allows the opening for reasoning.