Entry tags:
Linear algebra - Rasmussen, Wawro, Zandieh 2015
Rasmussen, C., Wawro, M. and Zandieh, M. (2015). Examining individual and collective level mathematical progress. Educational Studies in Mathematics, 88, 259-281.
The authors refer to Cobb and Yackel’s (1996) “emergent perspective and accompanying interpretive framework” (p. 259) as an early effort at integrating different theoretical perspectives to understand teaching and learning. The two theoretical perspectives referred to are constructivism (individual cognitive) and symbolic interactionism (sociocultural), citing von Glasersfeld and Blumer respectively. That framework already included a social perspective and an individual perspective, where classroom mathematical practices (social) were paired with mathematical conceptions and activity (individual). The authors split each of those two into two, resulting in four perspectives, two social and two individual.
The questions which can be posed for these four constructs are (p. 261):
The focus of analysis was video recordings of small group and whole class discussions in a linear algebra class of students in their second or third year of university in engineering or science degrees. In the paper there is particular focus on the concept of linear dependence. One group of five students was a primary focus group. For each of the four constructs, different modes of analysis were used. An argument of the paper is that using different modes of analysis on the same body of data using these four lenses (for example) can lead to a rich synthesized analysis.
The modes of analysis for the four constructs were:
The paper draws the analysis together by suggesting that analyzing the same greater body of data through four different modes of analysis related to four different constructs related to (individual and social) learning at the very least provides nuance to analysis by looking at one thing in four ways. However the four constructs are interrelated and analysis through one lens can augment analysis through another. One can look for correlations – “are different participation patterns correlated with different mathematical growth trajectories?”- or consistency – “In what ways are particular classroom mathematical practices consistent … with various disciplinary practices?” and so forth (p. 278-279). They cite Prediger et al. (2008) as calling for connecting and coordinating theoretical approaches to gain “explanatory, descriptive, or prescriptive power” (p. 278) and offer their framework and associated modes of analysis as an answer to this call. The paper frames itself as a first step in trying out this approach of coordinating analyses and anticipates further work.
I found this paper quite hard to read the first time, it is very dense with a lot of fine grained theory and data analysis. Upon multiple rereads, it all became clear and I can see the power in this framework where quite different analyses are carried out on the same body of data from quite different points of view, yet are tied together through the individual cognitive and sociocultural views of learning. In order for the data to be suitable for analysis it would have to involve a great deal of social interaction, such as these videotaped classroom and small group discussions. That makes me think about how I could (with my data which is less likely to be of that form) create a similar framework of interrelated constructs analysed through different lenses and resulting in a coordinated and nuanced piece of research work.
This is one of those papers you have to come back to multiple times to get full value.
Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understanding and opinions of Torquetum only and could contain errors, misunderstandings or subjective views.
The authors refer to Cobb and Yackel’s (1996) “emergent perspective and accompanying interpretive framework” (p. 259) as an early effort at integrating different theoretical perspectives to understand teaching and learning. The two theoretical perspectives referred to are constructivism (individual cognitive) and symbolic interactionism (sociocultural), citing von Glasersfeld and Blumer respectively. That framework already included a social perspective and an individual perspective, where classroom mathematical practices (social) were paired with mathematical conceptions and activity (individual). The authors split each of those two into two, resulting in four perspectives, two social and two individual.
- Classroom social practices (social) become
- Disciplinary practices and
- Classroom mathematical practices
- Mathematical conceptions and activity (individual) becomes
- Participation in mathematical activity and
- Mathematical conceptions
The questions which can be posed for these four constructs are (p. 261):
- Disciplinary practices: What is the mathematical progress of the classroom community in terms of the disciplinary practices of mathematics?
- Classroom mathematical practices: What are the normative ways of reasoning that emerge in a particular classroom?
- Participation in mathematical activity: How do individual students contribute to mathematical progress that occurs across small group and whole class settings?
- Mathematical conceptions: What conceptions do individual students bring to bear in their mathematical work?
The focus of analysis was video recordings of small group and whole class discussions in a linear algebra class of students in their second or third year of university in engineering or science degrees. In the paper there is particular focus on the concept of linear dependence. One group of five students was a primary focus group. For each of the four constructs, different modes of analysis were used. An argument of the paper is that using different modes of analysis on the same body of data using these four lenses (for example) can lead to a rich synthesized analysis.
The modes of analysis for the four constructs were:
- Disciplinary practice: Recognise that the disciplinary practices of professional mathematicians take the form of defining, algorithmatizing, symbolizing and theoremizing [sic]. Employ grounded approach on analysis of student data to see if categories of student activity relate to these professional categories. (When I return to doing stuff on proof I should follow this “theoremizing” strand to see where it leads.)
- Classroom mathematical practices: Toulmin’s (1958) model of argumentation; the data, the claim, the warrant. I saw Toulmin’s work being cited quite a bit when I was reading about proof. Because of that I ended up citing it myself in my 2017 SASEE paper. To see in detail how this paper used Toulmin’s model will be a key issue bringing me back to read this paper again. [Toulmin: The uses of argument]
- Participation in mathematical activity: Two frameworks of Krummheuer (2007, 2011) are used to identify roles in a conversation. The production design construct of mathematical conversation identifies authors, relayers, ghostees and spokesmen; the recipient design contract identifies conversation partners, co-hearers, over-hearers and eavesdroppers.
- Mathematical conceptions: The topic being focused on was linear dependence; Wawro and Plaxco (2013) describe four ways in which students conceptualise this topic and that framework is drawn upon here; travel, geometric, vector algebraic, matrix algebraic.
The paper draws the analysis together by suggesting that analyzing the same greater body of data through four different modes of analysis related to four different constructs related to (individual and social) learning at the very least provides nuance to analysis by looking at one thing in four ways. However the four constructs are interrelated and analysis through one lens can augment analysis through another. One can look for correlations – “are different participation patterns correlated with different mathematical growth trajectories?”- or consistency – “In what ways are particular classroom mathematical practices consistent … with various disciplinary practices?” and so forth (p. 278-279). They cite Prediger et al. (2008) as calling for connecting and coordinating theoretical approaches to gain “explanatory, descriptive, or prescriptive power” (p. 278) and offer their framework and associated modes of analysis as an answer to this call. The paper frames itself as a first step in trying out this approach of coordinating analyses and anticipates further work.
I found this paper quite hard to read the first time, it is very dense with a lot of fine grained theory and data analysis. Upon multiple rereads, it all became clear and I can see the power in this framework where quite different analyses are carried out on the same body of data from quite different points of view, yet are tied together through the individual cognitive and sociocultural views of learning. In order for the data to be suitable for analysis it would have to involve a great deal of social interaction, such as these videotaped classroom and small group discussions. That makes me think about how I could (with my data which is less likely to be of that form) create a similar framework of interrelated constructs analysed through different lenses and resulting in a coordinated and nuanced piece of research work.
This is one of those papers you have to come back to multiple times to get full value.
Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understanding and opinions of Torquetum only and could contain errors, misunderstandings or subjective views.