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Learning to listen: From the RhindPapyrus to classroom practice Abraham Arcavi The Hebrew University of Jerusalem Masami Isoda CRICED – University of Tsukuba

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Policy Measurement purposes (decision making, accountability, comparisons) Influence practice Intended Implemented Achieved

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Policy Measurement purposes (decision making, accountability, comparisons) Influence practice Intended Implemented Achieved

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Policy Learning Almost any teacher- student interaction involves assessment of what students know and understand

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For students For teachers

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Learning Learning to listen

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What is it?How may it be learned?

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“…students proceed in personally reasonable and productive ways.” Confrey, J.: 1991, ‘Learning to Listen: A student's understanding of powers of ten’, in E. von Glasersfeld (ed.), Radical Constructivism in Mathematics Education, Kluwer, pp. 111-138. Basic assumption 1:

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What is ‘listening to students’? (beyond mere physiology and beyond passiveness) Giving careful attention to what students say and do, trying to understand it, its possible sources and its entailments. It should include the following components: -Detecting and creating opportunities in which students are likely to engage in expressing freely their mathematical ideas; -questioning students in order to uncover the essence and sources of their ideas; -analyzing what one hears (sometimes in consultation with peers) making the enormous intellectual effort to take the ‘other’s perspective’ in order to understand it on its own merits; and - deciding in which ways to productively integrate students’ ideas.

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Why is it important? - Constructivism - Caring, receptive, empathic conversations - Internalized technique for learning and for interpersonal relationships - For ourselves, the listeners “Thinking ourselves into other persons leads us to reflect about our own relationship to mathematics” (Jahnke, 1994, p. 155). Jahnke, H.N.: 1994, ‘The historical dimension of mathematical understanding – Objectifying the subjective’, in J.P. da Ponte & J.F. Matos (Eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education, vol. 1, Lisbon, Portugal, pp. 139-156.

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Why is it important? - Constructivism - Caring, receptive, empathic conversations - Internalized technique for learning and for interpersonal relationships - For ourselves, the listeners “Thinking ourselves into other persons leads us to reflect about our own relationship to mathematics” (Jahnke, 1994, p. 155). “Confessions” of two mathematicians

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The challenges of ‘listening’ - “Packaged” knowledge “I have observed, not only with other people but also with myself…that sources of insight can be clogged by automatisms. One finally masters an activity so perfectly that the question of how and why is not asked any more, cannot be asked any more, and is not even understood any more as a meaningful and relevant question.” (Freudenthal, 1983, p. 469). “Highly practiced cognitive and perceptual processes become automatized so there is nothing in memory for experts to “replay”, verbalize, and reflect upon” (Nathan & Petrosino, 2003, p. 907). Challenge: Unpacking, unclogging, even “unlearning”

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The challenges of ‘listening’ - “Packaged” knowledge - “Decentering” capabilities “Making sense of children ideas is not so easy. Children use their own words and their own frames in ways that do not necessarily map into the teacher’s ways of thinking.” “The ability to hear what children are saying transcend disposition, aural acuity, and knowledge, although it also depends on all of these.” (Ball, 1993, p.378) Challenge: the capacity to adopt the other’s perspective, to ‘wear her conceptual spectacles’ (by keeping away as much as possible our own perspectives), to test in iterative cycles our understanding of what we hear, and possibly to pursue it and apply it for a while.

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The challenges of ‘listening’ - “Packaged” knowledge - “Decentering” capabilities - Different ways of listening Challenge: “evaluative” vs. “attentive” listening “Evaluative” listening: listening against the background of an expected answer. It implies a virtual ‘measurement’ of the ‘distance’ between the student present state of knowledge and the desirable goal, providing straightforward feedback and applying subsequent ‘fixing’ strategies. Such listening usually disregards the students’ thinking,

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The challenges of ‘listening’ - “Packaged” knowledge - “Decentering” capabilities - Different ways of listening - Timing Challenge: reflecting in real time.

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Basic Assumption 2: Productive listening is learnable Research question(s) What kind of experiences should be orchestrated in order to develop desirable listening capabilities? What kind of “curriculum for professional learning” and “pedagogy for professional development” should be developed? How such a curriculum may work in teacher courses and to what extent the goal of learning to listen can be attained? A first answer: role models as in lesson study

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The rationale for our approach In learning history of mathematics and in learning to listen one has to learn to interpret This work is about connecting these two types of learning

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Basic assumptions 3: a)In order to fully understand the ideas behind a historical (mathematical) source we need a similar kind of unpacking and decentering needed for listening to students; b)Suitability of the historical context (respectability of source, development of “hermeneutic tools”)

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What "hermeneutic tools"? - parsing the source - posing questions to oneself (or to a peer) : What is really written? Why did the author write in such a way? What are the hidden assumptions? If this text says A, and A entails B – where is B in the text? This questioning may lead to adopt the ‘writers’ perspective’; - paraphrasing parts of the text in our words and notations; - summarizing partial understandings, locating and verbalizing what it is still to be clarified; - contrasting different pieces for coherence. This practice may help develop an understanding which needs to be re-confirmed with a recursive process (e.g. applying our understandings to similar texts, examples or problems).

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A special way of work was designed and implemented to support the understanding of certain type of primary sources from the history of mathematics. Conditions: 1)Selection of sources 2)Design and implementation

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EXAMPLES

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First stage: The following text (Figure 1) from the Rhind Papyrus presents a calculation, in Hieratic and Hieroglyphics, performed during the solution to problem # 74. 1- On the basis of the initial translation to modern notation, indicate what each of the Hieroglyphic signs may mean. 2- Complete the blank spaces. 3- Explain what calculation was performed in this extract.

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First stage: The following text (Figure 1) from the Rhind Papyrus presents a calculation, in Hieratic and Hieroglyphics, performed during the solution to problem # 74. 1- On the basis of the initial translation to modern notation, indicate what each of the Hieroglyphic signs may mean. 2- Complete the blank spaces. 3- Explain what calculation was performed in this extract. 5602 11204 19607

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Second stage: The following presents another calculation, performed during the solution to problem # 52 Given your experience of the previous step, what questions would you pose to the new text?

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Second stage (cont.): - On the basis of the initial translation from to modern notation, indicate what each of the signs may mean. - Complete the blank spaces. - Explain what calculations were performed in this case, and what is the meaning of the slash marks? - Calculate 13x27 by the Egyptian method. - Can one multiply any pair of numbers by this method?

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Third stage: The following is the solution to Problem # 24 (Peet, 1970). The omissions are for the purpose of this task.

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- Generate questions whose answers would help you understand the solution process to the linear equation. - Write, in modern notation, the equation corresponding to the first sentence, and solve it. Questions 3-9 are aimed at helping to understand the solution method

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- Reproduce and summarize the method of solution and explain it. - Write down the solution of the problem, as it would have appeared in the Papyrus (but in modern notation), if the first trial number had been 14 instead of 7. - Problem #25 in the Papyrus is “A quantity whose half is added to it becomes 16”. Solve the problem as you would today and using the Egyptian method (as you think it would appear in the Papyrus, but using modern notation).

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- ‘dictionary’ translations of notations - description of a calculation (surface and deep structure) - solution with our method - parsing of the text, concentrating on ‘local’ understanding of small parts - ‘pasting pieces together’ towards a global understanding - applying the Egyptian method to a new problem - clarifying previous knowledge, hidden assumptions, conditions of applicability - investigating mathematical properties, generality. Supporting questions

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A workshop - Participants - Duration - Components Introduction Work on the Egyptian source Second source Questionnaire I Pedagogical component

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Task borrowed from Even & Wallach (2004). A., a fifth grade teacher, wanted to assess whether her students know how to find the whole when a part is given. She administered her students a quiz that included the following problem: “3/5 of a number is 12, what is the number explain your solution.” This is what Ron wrote: “12 * 2 = 24, 24:6=4, 24-4=20”. - Is Ron’s solution correct? - If you were Ron’s teacher, what would be your assessment of Ron’s knowledge?”

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A workshop - Participants - Duration - Components Introduction Work on the Egyptian source Second source Questionnaire I Pedagogical component Discussion of connections Questionnaire II Summary – explicating goals

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What did we learn from the workshop? Is the approach feasible? Learning about ‘teaching practice’ Learning to ask “Evaluative listening” Is interpreting texts = listening to students? What about experienced teachers? Development of “tools for listening”

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12 24 12 24 12 4

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(a) Decentering aimed at making sense of the other’s perspective may rely on tools we, as interpreters, bring to the situation, in order to mediate the construction of our understandings; and (b) The kinds of tools we bring can be of very different types, ranging from those which can be attributable to the students to those clearly beyond their reach – in both cases, tools may be helpful to the interpreter. However, some tools may be mistakenly misattributed to students.

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Final words Existence proof: It is possible to educate towards listening, and there is more than one approach to do so. History is promising. Requires careful design It may have unexpected nuances and complexities A curriculum of this kind should be embedded within a conglomerate of teacher development activities, discussions and reflections coherently geared towards the development of a web of interconnected beliefs capable of sustaining attentive listening.

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どうもありがとう Terima kasih 十分感激 Thank you תודה רבה Muito obrigado Muchas gracias Grazie Спасибо Maraming salamat

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