"...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
"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.
"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."
"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..."
"...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..."
"...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)."
"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.
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.
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:
for mathematics (movement back and forth) classroom activity (movement back and forth) sociomathematical knowledge
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.
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.
Figure 3 embedded on hybridalgebra.wikispaces.com- 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 5 (lines 5 and 9), the Equality Property6 (lines 8 and 13) and the Distributive Property of Multiplication Over Addition 7 (line 10)."
"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!
"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."
"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 t 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."
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.