Friday, July 31, 2009

Math Lessons Collaboration Daily 08/01/2009

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Wednesday, July 29, 2009

Math Lessons Collaboration Daily 07/30/2009

Posted from Diigo. The rest of my favorite links are here.

Tuesday, July 28, 2009

Math Lessons Collaboration Daily 07/29/2009

  • tags: no_tag

    • pg 258 "In collaborative discussions, the teacher's comments direct students not only to understand what others are saying, but also to respond to, extend, or connect to those ideas. For example, the teacher can ask, "Does anyone want to build on what sally just said?"...However, the teacher can encourage students to  work directly with one another's ideas and can take steps to foster such interactions."


      pg 258-9 "To get students to listen to, comprehend, and extend other's ideas, teachers must ask them to do this consistently and not only in a whole-class format. For example, during a group work, the teacher can actively support collaborative discussions by engagin all students at the table in the conversation...the teacher responded only to "group questions"  or called on specific students to ask the question to make sure that everyone was in agreement with the necessity for a question and what the topic should be.



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    • pg 258 "We capture differences in these kinds of discussions by talking about three key aspects of pedagogy: positing students for discussion, managing wrong answers, and connecting and linking across ideas."


      Annecdotal evidence from observation and analysis of lesson film.

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    • pg 257 "We focus on two kinds of classroom discussions: sharing discussions and collaborative discussions. In sharing discussions, students share their answers and teachers value their contributions... In collaborative discussions, students also share their ideas, but in addition, the build on the thinking of their classmates' responses, take up classmates' ideas, and work with these ideas explicitly to extend the line of thinking that emanates. "

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    • pg 88 "Questions such as the following might orient this work.
      * What is the communicative history of a particular activity structure (e.g.,
      science talk) in a given classroom?
      * How is disciplinary knowledge interactionally accomplished through
      discourse within this activity structure?
      * What do individual children discursively contribute to this collective
      activity, and how do their contributions change over time?
      * How do the voices of individual children, reflective in part of community
      membership, intermingle over the course of the construction of classroom

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    • pg 76 "When Michaels asked her why she didn't explain her solution processes in the first place, Elizabeth responded", I didn't understand your question" (Michaels & O'Connor, 1990, p. 16)."


      Teachers must scaffold/support metacognitive questions so that students become aware of their thinking patterns. Students do not automatically understand how to express their thinking verbally or in written form so teachers must, "Embed Communities of Discourse"in their instruction.

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    • jjpg 72 "Classroom Discourses and Educational Reform
      Educators working across disciplinary domains have begun to recognize that educational reform requires addressing traditiona classroom discursive practices. Inquiry learning, for instance, cannot occur as long as teachers and students communicate largely in ways reflective of direct instruction, where the teacher and the classroom texts hold the key to knowledge in the form of the "right" answer. Many educational reform efforts that have made discourse a  centerpiece of reform have done so from a sociocognitive or sociocultural theoretical framework. Examples of such efforts from three academic discipline areas-reading and language arts, science, and mathematics illustrate how research on discourse has played a key role in attempts at improving classroom teaching and learning."


      Discourse must change if instruction is to change.

      Key component of sociocognitive framework is that students communicate with each other around the construction of understanding/content.

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    • "In my own work on narrative discourse and classrooml earning( see Hicks, 1993), I have argued that narrative discourses should be examined in light of the speech activities in which participants are engaged. In this sense, narrative can be explored as a family of discourse genres in which children and teachers construct extended oral
      or written texts that order, describe, explain, or emplot events, both real and fictional."
      pg 70 "Studies of children's language use and classroom learning have suggested a
      relationship between such extended talk about events and children's successful
      participation in school tasks(Snow, in press). In large part, such studies have
      connected children's narrative performances with their literacy learning. Since
      oral narrative genres represent extended discourse in which the speaker constructs
      a text as opposed to a single utterance, studies of children's oral
      narratives have been viewed as a possible "window" onto their classroom
      literacy learning. Such work has often been focused on sociolinguistic differences
      in the narratives constructed by children from differing communities.
      Recall that Heath's ethnographic study of communities' ways with words
      revealed important differences in narrative socialization. In subsequent studies
      of children's classroom narratives, Michaels (1981) found cultural differences
      that supported Heath's work. The Sharing Time narratives of Black children
      were rooted more in an "oral" discourse style, where events were connected
      through linguistic devices like repetition, such as one might find in poetry.
      The SharingT ime narratives of White children tended to be more essayist
      or "literate"in nature, in that a single topic was chosen and events connected
      to the topic were related in a sequential manner. Researchers like Sarah
      Michaels have concludedt hat differing forms of narrative discourse may
      provide children with differential access to literacy. As Scollon and Scollon
      (1981) note in a chapter titled "The L iterate T wo Year Old," preschool children
      in many middle-class families may be literate long before they are able to
      read and write."


      Implications for dealing with different student populations and needs to address different cultural learning styles.

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    • Classroom Genres of Discourse:
      pg 69 Monologic form of talk. A child in a mathematics classroom may be asked to elaborate on how she arrived at her answer to a problem... The speaker is given the floor for a much longer period of time. The "rules of the game"-the participant frameworks that define social roles and obligations-have shifted somewhat so that the speaker no longer expects an immediate response.
      pg 70 Narrative discourses, oral and written texts representative of a series of connected events, may be another overarching discourse genre in the classroom, one with many variations in both form and function...Two examples that come to mind are (a) Morning Circle (or First Circle) and Sharing Time, where children construct narratives of personal experience (Dorr-Bremme1, 992;Kantor, Green, Bradley, & Lin, 1992; Michaels, 1981), and (b) Journal Writing, where children
      typically write journal entries grounded in fictional or personally experienced events (Dyson, 1989, 1993; Hicks & Kanevsky, 1992)."

      pg 70 "In my own work on narrative discourse and classrooml earning( see Hicks, 1993), I have argued that narrative discourses should be examined in light of the speech activities in which participants are engaged. In this sense, narrative can be explored as a family of discourse genres in which children and teachers construct extended oral or written texts that order, describe, explain, or employ events, both real and fictional."


      Different forms of discourse in the classroom. Focus on discourse (especially in younger students) that is not fully content related. A major difference is that the context of the learning for younger students is more closely tied to the general understandings at the secondary level discourse can move to more content 'heavy'/specific, but it should be noted that discourse/learning is still a social construct and context is still critical. A scaffolded approach must be made, processes identified, routines developed, long-term focus on helping students understand content related/specific discourse that is less connected to personal/fiction constructs.

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      • pg 67

        "In his article "Reevaluating the IRF Sequence," Gordon Wells (1993)
        describes how one classroom teacher moved between direct modes of instruction
        involving a triadic form of discourse with explicit evaluation and discourse
        that resembled more what Tharpe and Gallimore (1988) refer to as
        an instructional conversation


        Reference for Wells: Wells, G. (1993). Reevaluating the IRF sequence: A proposal for the articulation of theories of activity and discourse for the analysis of teaching and learning in the
        classroom. Linguistics and Education, 5, 1-37.


        Need definition of triadic discourse (between teacher-student-student?)


        Reference for Tharpe and Gallamore: Tharpe, R. G., & Gallimore, R. (1988). Rousing minds to life: Teaching, learning, and schooling in social context. New York: Cambridge University Press.

        Definition of instructional conversation:

        According to Cazden (1988, p. 54), instructional conversation is "talk in which ideas are explored rather than answers to teachers' test questions provided and evaluated." Goldenberg provides five critical features of this type of teacher-student interaction:

        • It is interesting and engaging.
        • It is about an idea or a concept that has meaning and relevance for students.
        • It has a focus that, while it may shift as the discussion evolves, remains discernible throughout.
        • There is a high level of participation, without undue domination by any one individual, particularly the teacher.
        • Student engage in extended discussions -- conversations -- with the teacher and among themselves.
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    • pg 67

      "the IRE/IRF sequence has come to be associated with broader teaching practices or philosophies, such as the direct instruction of skills or the elicitation of so-called known-information questions, questions where the teacher is seeking the "right answer." The IRE sequence has, for some educational researchers, come to be associated with a "skill and drill" mode of instruction, and reform efforts have therefore made such modes of communication one of their primary targets of change."



      Further evidence of lower-level discourse expectations.

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    • pg 66
      "IRE/IRFT: he Unmarked Case?
      Across the literature on classroom discourse, certainly the most robust
      form of talk documented has been that of the Initiation-Response-Evaluation
      (Cazden, 1988; Mehan,1 979) or Initiation-Response-Follow-up (Sinclair &
      Coulthard, 1975; Wells, 1993) sequence. This genre has come to be viewed
      as what linguists call the unmarked case; it constitutes somewhat of a norm.
      Perhaps the robustness of the IRE/IRF sequence across classroom settings
      can be related back to the example of mother-child interaction shown in
      Example 4 (lower-level, fill in the blank/rote memory kind of response) . There, as the more capable adult in that example assumed an instructional role, she provided conversational slots so that the child could successfully participate in joint book reading. The provision of such slots or  openings for learners may be a means through which caretakers, and teachers, provide access to discourses and forms of knowledge beyond the child's independent means.

      The IRE/IRF organization of classroom lessons has been documented
      largely for group discussion activities in which the teacher asks a question,
      a child (or a group of children) responds, and the teacher follows through
      with a comment, often evaluative in nature. The following example from
      Courtney Cazden's own teaching illustrates the typical triadic structure of
      classroom lessons. In this example, the teacher has been eliciting children's
      birthplaces as a means of helping them learn about geography, distance, and
      family origins. One student, Prenda, has been called on to identify her birthplace
      and then to find it on a map."


      Confirms my experiences in the classroom that teachers rely on very low level, one word responses from students as the majority of interactions.

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    • pg 62

      "This concludes my discussion of theoretical perspectives on discourse and
      learning. I might summarize this section by stressing that discourse is an
      inherently social construct that mediates, indeed partly constitutes, the teaching
      and learning that take place in classrooms. Through meaningful classroom
      activity, children appropriate the discourses that situationally define "what
      counts" as knowing within disciplines. In the following section, I turn to
      current research on classroom discourse. I explore prominent classroom
      genres, such as the Initiation-Response-Evaluation/Initiation-Response-Follow-
      up instructional sequence; efforts at educational reform tied to discourse;
      the "embeddedness" of classroom discourses in community-based discourses;
      and the heterogeneous discourses that are woven into one text or turn of talk.
      Each of these topics is elaborated through reference to work of current
      researchers in the field. Because of the breadth of this field, I cannot hope
      to cover all of the research that would fall under a given topic. Rather, I
      provide exemplars that will, I hope, enrich readers' understanding of the
      research topics currently being addressed within the field of classroom


      Nice synopsis of the discourse conversation so far.

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    • pg 59

      "How do teachers teach and children learn these more formal academic
      discourses?  There have been some efforts to incorporate the explicit teaching
      of academic genres into literacy instruction (see Reid, 1987, for a discussion of
      the controversy surrounding "genre instruction"). However, a sociocognitive
      perspective on discourse would suggest that children learn academic discourses
      through their repeated participation in meaningful social activity (Rogoff, 1990). Classroom teachers set the stage for this learning process by providing the discursive" slots"t hat enablen ovice learnerst o participatein disciplinaryp ractices.T his can occur withing roupd iscussionsi n which the teachero rchestratesc lassroom discourse (NCTM, 1991, p. 35), or it can occur in small group or one-on-one conferences."


      Difficult to teach the content literacies in generic form as the learning is social and must be connected to the learning before it is internalized. Further evidence/necessity that discouse must be established/developed/supported in the content area.

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    • pg 58

      "Literacy is generally thought of as the ability to read and write. However,
      some theorists have extended the meaning of literacy to encompass more
      than just the ability to read and write. Full literacy, according to some, entails
      mastery of secondary or formal institutional, often academic discourses (Gee,
      1989). These secondary discourses typically involve ways of describing,
      explaining, and questioning that are dissimilar to "ordinary" conversation."


      Great rationale for the inclusion of dialog in our conversations of literacy and nice description that content literacy is more that conversation and contains skills/concepts/components that need to be taught/supported/developed.


      Implications for teachers: scaffolding/gradual release/experiences that include multimodal forms of communication ...

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    • pg 58

      "Similarly, some theorists would suggest that "learning science" is a process that is framed by discourse genres, forms of activity, and ways of establishing semantic
      links among events, objects, and persons (see Lemke, 1988, 1990; Rosebery,
      Warren,& Conant, 1992). Other researchers have explored similar issues
      with respect to mathematics instruction (Ball, 1991, 1993; Bill, Leer, Reams,
      1992; Hiebert & Wearne, 1993; Lampert, 1990; Spanos, Rhodes, Corasaniti
      Dale, & Crandall, 1988)."


      Connections between interactions in math and science classes are closely related.

      Also, several references to consider:

      Ball, D. (1991, November). What's all this talk about "discourse"? Arithmetic Teacher; pp. 44-48.
      Ball, D. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 93, 373-397.

      Lampert, M. (1990). When the problem is not the question and the solution is not
      the answer: Mathematical knowing and teaching. American Educational Research
      Journal, 27, 29-63

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    • pg 55

      "and teacher and student interactions in classroom settings have drawn upon sociocognitive theory to explain how children learn in what Vygotsky termed the
      "zone of proximal development" (Vygotsky, 1978; see also Rogoff & Wertsch,
      1984). A central metaphor that has emerged from this work is that of scaffolding
      (Wood, Bruner, & Ross, 1976)."


      Important for classroom practice to recognize the necessity of scaffolding and creating experiences that push students development but don't move too quickly for students to garner experience/cognitive benefit.

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    • pg 53

      "Discourses, some of which may be conflicting
      identities for them. For instance, my own working-class roots can be in
      conflict with the Discourses that I must assume for the purpose of being
      an academic; similarily, academic Discourses may be in conflict with the
      Discourses that are associated with my being a woman (Ellsworth, 1989;
      Hollingsworth1, 994).
      It is important to add that academic discourses also embody such ideologies;
      they are also Discourses. Thus, learning to "talk science" or "talk math"
      involves more than just learning a set of linguistic forms; it also involves
      learning beliefs and values (Lampert, 1990; Lemke, 1990; Yackel et al.,
      Gee (1989, 1990) and Lemke (1990) both point out that the ideologies
      embodied in such academic discourses maybe in conflict with some children's
      home and community discourses. Minimally, these academic discourses may
      be unfamiliar to children who have not experienced at home what Scollon
      and Scollon (1981) term essayist literacy."

      Important to understand that discourse is not separated from environment, culture, experience.

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    • Toward an operational definition of discourse from the cognitive perspective: "For reasons that will become more clear as this review progresses, the term discourse implies communication that is socially situated and that sustains social "positionings": relations between participants in face-to-face interaction or between author and reader in written texts."


      "The term discourse implies a dialectic of both linguistic form and social
      communicative practices. One can talk of discourse in terms of oral and
      writtent exts that can be examined after the fact and socially situated practices
      that are constructed in moment-to-moment interaction ( Fairclough, 1992;
      Gee, Michaels, & O'Connor, 1992). Thus, use of the term discourse implies
      a decision about how classroom communicationis to be theoretically positioned
      in research on teaching and learning"

  • Discourse, Learning, and TeachingDeborah Hicks

    tags: discourse, dialog, math

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Monday, July 27, 2009

Math Lessons Collaboration Daily 07/28/2009

  • tags: discourse, math, dialog

  • tags: no_tag

    • pg 2

      "...core recommendations of the reform movement are carried out: students use visual models and manipulatives; they work together in groups; they are expected to focus on the process of doing mathematics rather than on just obtaining an answer; they are expected to discuss their understadnings and to communicate their findings; they work on open-ended problems that often require more than 1 class period to solve; and they are asked to make conjectures, to justify their thinking, to assess their knowledge, and to take a degree of responsibility for their learning. In short, many of the visible elements of reform are present in this classroom and provide a comparatively rich environment for learning mathematics."

       Nice description of components of a reform oriented classroom, but more specifically the processes that are critical to developing a focus on dialog in the classroom. In a middle school setting, but nothing prohibits this approach in the secondary setting.

      Finally, not strong research, much self-report and qualitative in nature, but can still access the processes/strategies/focus/routines

  • Research gathered from studeis around discourse in mathematics. Working document

    tags: discourse, dialog, math, secondary

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  • tags: no_tag

    • References for further research:

      Manouchehri, A., & Enderson, M. C. (1999). Promoting mathematical discourse: Learning from classroom examples.


      Mathematics Teaching in the Middle School, 4 , 216–222.

      Langer, J. A. (2001). Beating the odds: Teaching middle and high school students to read and write well.


      American Educational Research Journal, 38 , 837–880.

      Pugalee, D. K. (1999). Constructing a model of mathematical literacy.


      The Clearing House, 73 (1), 19–22.

  • tags: no_tag

    • pg 16

      "Our knowledge function analysis of participants’ discourse indicates a gap in linguistic functions characteristic of teacher talk, as compared with student talk—with student talk reflecting lower-level knowledge structures, even in the presence of teacher’s discourse that illustrated varied, higher-level knowledge structures. These results suggest that the features of teacher discourse do not automatically transfer to student discourse through class discussion. Yet, when students were put in the position of “teaching,” they did demonstrate the construction of higher-level knowledge structures that were present in teacher discourse."

      Focus for results is to create opportunities for students to speak to students using the language of the teacher, the teacher to model but not expect that to do much unless coupled with student speaking opportunities.

      pg 16

      "The study suggests that instructional design in math education would be best served by systematically integrating math thinking and math talking at all levels of knowledge structures. Also, since conceptual understanding and procedural knowledge (representing the classification and principles knowledge structures) are the very foundation of mathematical reasoning, instructional design plans would benefit from a widening of the range of discourse functions to especially include those associated with higher-level knowledge structures. Teaching strategies should promote such discourse by students. To achieve this goal, teachers need to play the role of both a mathematician and a mentor—to do math but also to “talk” math as a way to model for students the way they should talk math."

      Further rationale for study/project

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    • pg 13

      "For instance, right after the instance in Vignette 9 (when Ms G tried to push students to articulate principles but did not follow through), two more equations were written on the chalkboard. This time, the students were asked to work as individuals, and the class was absolutely silent. When students were placed in a position to talk as a “teacher,” though, students did display higher-level knowledge structures. This transformation of student talk was evident in Vignette 6. Perhaps placing students in a teacher role helped them think more intensively in terms of their classroom audience, and therefore move towards..."

      pg 14

      "...higher-level knowledge structures in their discourse. (What does the audience need to know beyond the sequence of steps? How can I connect the steps to the principles? How can I help them connect what they don’t know with what they do know?). Perhaps allowing students to speak from a teacher’s role in a peer-to-peer..."

      p 15

      "...context provides them with an opportunity to reflect on their knowledge structures. “Think aloud” protocols, in which students are asked to articulate their knowledge in an explicit fashion, have been shown to help students develop metacognition, which is so crucial to math learning (Davey, 1983)."

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    • pg 9

      "The following examples show how student discourse was limited in expressing higherlevel knowledge structures when conceptual understanding was the goal of the lesson. Specifically, our analysis suggests that the classification knowledge structure occurred frequently, as illustrated in Vignettes 4 and 5, but such discourse was usually produced in an elaborated fashion by the teacher only, while student discourse expressing theory aspect structures was limited, if it existed at all.

      Though these data revealed that student talk was limited in expressing higher-level knowledge structures, we did find some examples of student discourse that


      demonstrated higher-level knowledge structures. These theoretic aspects of knowledge structures occurred in student discourse primarily when students acted as teachers themselves, as illustrated in Vignette 6

      Reinforces the need for students to speak to each other, not the teacher speaking for them or to them.



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    • In the field of communication in education, there have been numerous studies of oral discourse in the classroom (see Rubin, 2002).

      Rubin, D. L. (2002). Binocular vision for communication education. Communication Education, 51, 412–419.

      Many focus on “talk as learning” in the contexts of English, language arts and social studies (e.g., Nystrand, 1997; Nystrand, Gamorean, & Carbonaro, 1998). These studies have clearly identified connections between knowledge construction, communication, and students’ learning(e.g., Comstock, Rowell, & Bowers, 1995; Langer, 2001). Within the field of mathematics education, the role of communication in mathematics learning has also been extensively explored.While available studies suggest communication is one of the key processes in building understanding (Hiebert et al., 1997; MacGregor & Price, 1999; Manouchehri & Enderson, 1999; Monroe, 1996; Warfiel, 2003), most seem to focus on communication as a means to achieve the goal of acquisition of math content (e.g., Borasi, Siegel, & Fonzi, 1998) and not as an objective in itself. Additionally, most of the studies focus on elementary math settings and do not fully explore secondary mathematics classrooms.

      Most of the study in the field has been at the elementary level, indicating a need for more study at the secondary level.






  • tags: no_tag

    • The language socialization concept provides useful insight into the relationship between the learning of math content and the acquisition of math language. Thus, following Ochs, we propose an activity model as follows:


      language of/


      language of/

      language of/


      for mathematics (movement back and forth) classroom activity (movement back and forth) sociomathematical knowledge

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    • The National Communication Association promotes the role of communication as a vehicle for “creating meaning, influencing thought, and making decisions” (National Communication Association, 1998). Consistent with that view, the National Council of Teachers of Mathematics in their Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) make communication an important objective in the teaching and learning of mathematics by including the following communication standard: “Instructional programs from prekindergarten through grade 12 should enable all students to organize and consolidate their mathematical thinking through communication, and communicate their mathematical thinking coherently and clearly to peers, teachers, and others . . .” (p. 348).

      Rationale for the importance of discourse in the mathematics classroom.

  • tags: discourse, math, dialog

  • tags: no_tag

    • Student quote "and ``r'' equals ten on tan of theta take tan of alpha"  for "r = 10/(tan theta - tan alpha)"

       A similar situation is demonstrated in the second example where the grammar of mathematical symbolism allows for ellipsis of the Operative process of multiplication together with brackets to indicate the grouping of nuclear configurations. When verbalized, however, there is ambiguity to the meaning of this linguistic statement.

      EDITORIAL:  Allowing students to use imprecise language and using imprecise language ourselves contributes to this laxity of understanding.







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    • Figure 3 embedded on ohallaran_figure_3

      "For example, the mathematical definitions and properties that have been assumed in the consequential relations in Fig. 5 include the definition of the tangent ratio1(lines 1 and 2), the Multiplication Property of Equality2 (lines 4, 6, 11), the definition of the Multiplicative Inverse3 (lines 4, 6, 11), the Addition Property of Equality4 (lines 5 and 9), the Definition of Subtraction (lines 5 and 9), the Equality Property6 (lines 8 and 13) and the Distributive Property of Multiplication Over Addition (line 10)."

  • tags: no_tag

    • "Structural condensation, whereby meaning is encoded in the most economical way possible, is achieved through devices such as:
      1. a specific rule of order for Operative processes (brackets, indices, multiplication/division and addition/subtraction)

      2. the possibility of alternate orderings through use of different forms of brackets

      3. ellipsis of the Operative process of multiplication

      4. exploitation of the resources of spatial graphology

      5. conventionalized symbolic forms"

      "For example, in the case of s(t) = ÿ16t^2 + 80t, the condensatory devices that have been employed include the rule of order so that multiplication, ÿ x 16 x t x t and  80 x t, occurs before addition of these two terms, ellipsis of the multiplication sign in -ÿ16t^2 and 80t, and use of spatial graphology so that t^2 means t x t. Other strategies involve the use of standard functional notation; for example, s(t) to mean the value of the function s at t."

      Great example of the meaning incurred in a simple function written symbolically!

  • tags: no_tag

    • "the impact of the multisemiotic nature of mathematics on classroom discourse to three main areas: first, the nature of the lexicogrammar of mathematical symbolism and its effect on the surrounding verbal discourse; secondly, some general features of mathematical pedagogical discourse arising from the multisemiotic nature of its makeup; and, thirdly, the shifts in meaning that result with movements between codes."

  • tags: no_tag

    •  "For example, Burgmeier, Boisen, and Larsen (1990, p. 83) give the mathematical description s(t) = ÿ16t 2 + 80t for the height of an arrow shot vertically into the air, where is the time in seconds. In this mathematical symbolic description, the complete pattern of the relationship between time and height of the arrow is encoded."

      The information that is presented in an equation or symbolic representation is/can be ver complete. That ability to present information in the computer era is getting stronger as diagrams and graphical representations become more dynamic.

      "In this respect, the visual display is not only limited in functionality, but also graphs and diagrams are usually only partial descriptions of the complete description encoded in the mathematical symbolic statements. In addition, there exists a possible misuse of diagrams, although Shin (1994) maintains that each of the above claims do not entirely warrant the lower status accorded to mathematical visual display. With the power of computers to dynamically display visual images, however, the status of this semiotic appears to be rapidly increasing."




  • tags: no_tag

    • Thus, the analysis of ``mathematical language'' must undertaken within the context of which it occurs; that is, in relation to its codeployment with mathematical symbolism and visual display. The analysis of the language of mathematics classrooms must necessarily be incomplete unless the contributions and interaction of the symbolism and visual display are taken into account."

      Reinforcing the multiple representation and having students work fluently with each form as well as work with back and forth with each form.

  • tags: no_tag

    • "Mathematics is multisemiotic because the linguistic, visual and symbolic semiotic systems differentially contribute to the meaning of the text."

      Multiple Representations, again.

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Saturday, July 25, 2009

Math Lessons Collaboration Daily 07/26/2009

  • "Staffed with about 60 full-time researchers, many of them Indians with PhDs from top universities in the United States, the center is at the cutting edge of Microsoft's R&D. It covers seven areas of research including mobility and cryptography."For now, the US is still the center of tech education. How long will people continue to come to the US for education? How can we change the tide of American students not choosing/being prepared for engineering/leadership

    tags: instruction

  • 280Slides is an online presenation tool. I played with it and created a quick presenation with embedded video really quickly. Just another PP tool for those who don't have access to Microsoft or want to do their presentations online.

    tags: web2.0, powerpoint, presentations, write2learn, write2pub

  • Through its Copyright Advisory Group, the Australian Ministerial Council on Education, Employment, Training and Youth Affairs (MCEETYA) has published a Creative Commons information pack online, a bundle of eight documents that distills the basics of CC licensing and the philosophy behind it. This pack is a great resource for educators and students, and we encourage you to use it in your schools by adapting it however you like. Read more: Under Creative Commons License: Attribution

    tags: creativecommons, copyright, publish

  • three characteristics of an alternative discourse of mathematics education are proposed. These are: that the learners make the mathematics; that mathematics involves thinking about problems; and that difference and individuality should be respected.

    tags: discourse, math

  • Great set of tips for using Twitter more effectively. I really like #3, because it's all about interaction in my opinion. If working with teachers on backchannel conversations/routines, then these tips should help the teachers understand how to support the process more effectively. Thanks Clif for sharing!

    tags: twitter, socialnetworking, write2learn

  • Having students start understanding number properties early is a significant aspect of developing algebraic reasoning. I really like that this game allows students to 'play'/explore different number combinations. I can even see using this game with older (3-5th) students to have them explore different characteristics of the problems, i.e. 3 + 4 = 7 the larger number goes at the end, 3 and 4 can be switched and still get 7 to introduce number properties in a concrete experiential way.

    tags: math, smartboard, iwb, addition

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Friday, July 24, 2009

Math Lessons Collaboration Daily 07/25/2009

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Saturday, July 18, 2009

Math Lessons Collaboration Daily 07/19/2009

  • Springnote allows you to create pages, to work on them together with your friends, and to share files. Springnote is also a great tool for group projects as it allows group members to easily collaborate. Advanced search, numerous templates, and 2GB of FREE File Storage are only few examples of how Springnote can help you. Of course, you already know that Springnote is an Internet service, meaning you can access it from anywhere anytime.

    tags: write2learn, web2.0, notetaking, wiki

Posted from Diigo. The rest of my favorite links are here.

Friday, July 17, 2009

Math Lessons Collaboration Daily 07/18/2009

  • Resources to find data on the internet

    tags: 21stCentury, literacy

  • In Intelligence Reframed Howard Gardner contends that "literacies, skills, and disciplines ought to be pursued as tools that allow us to enhance our understanding of important questions, topics, and themes." Today's readers become literate by learning to read the words and symbols in today's world and its antecedents. They analyze, compare, evaluate and interpret multiple representations from a variety of disciplines and subjects, including texts, photographs, artwork, and data. They learn to choose and modify their own communication based on the rhetorical situation. Point of view is created by the reader, the audience and the medium.

    tags: 21stCentury, literacy, comprehension, write2learn, write2pub

  • Top 50 iPhone Apps for Educators- Although you're not likely to see schools issuing an iPhone to every faculty and staff member, the fact is that the iPhone is a great tool for education. Whether you're a teacher, librarian, or other educator, there are a number of apps that can help you do your job better. Here, we'll take a look at 50 of these apps and what they can do for you.

    tags: iphone, applets, education

Posted from Diigo. The rest of my favorite links are here.

Tuesday, July 14, 2009

Math Lessons Collaboration Daily 07/15/2009

Posted from Diigo. The rest of my favorite links are here.

Monday, July 6, 2009

Math Lessons Collaboration Daily 07/07/2009

Posted from Diigo. The rest of my favorite links are here.