Knowledge Forum-using online knowledge forum as new context for mathematics learning

Moss & Beatty (2010): Knowledge building and mathematics

The purpose of Moss and Beatty’s study was to test the possibility of young students’ engagement in theory development in the context of mathematical problem solving. Three classes of grade four students were introduced to pre-algebra and participated in solving a series of difficult problems online.

This article introduced a learning context called Knowledge Forum. This is an online forum to provide an authentic context for collaborative problems solving and extended mathematical discussions while students worked on an open-ended school-based generalizing problems. In this context, the teacher posts a problem but then is ‘absent’ from online discussions. Instead students figure out solutions.

Knowledge Forum is based on Knowledge Building which is inquiry-based and centered around theory development. Knowledge Forum is based on two principles: democratization of knowledge and epistemic agency. Democratization of Knowledge counters traditional math classrooms were the teacher is the holder of knowledge and ‘high’ students’ contributions are favored compared to students who are struggling. Instead, Moss and Beatty found that all students, including ‘low’ or struggling students used Knowledge Forum. Low students who might not speak up in class contributed, albeit they spent more time signed in and reading online posts than introducing new mathematical generalizations. Epistemic agency describes the responsibility that the group assumes for ownership of ideas that are given in public life in this online environment. Moss and Beatty found that students took responsibility for contributing ideas, monitoring one another’s contributions, asking for clarification with the goal of moving theory forward.

In their discussion, Moss and Beatty make distinction between Knowledge Building and community of learners. A community of learners (Lambert, 1998 qtd. in Moss and Beatty, 2010) is when students engage in justifying and critiquing mathematical explanations with the goal of acting as an expert. In contrast, Knowledge Building favors theory development –that is the collaborative production of new ideas. Students spend less time justifying and critiquing and even when they do justify and critique the aim of such activity is the production of new ideas. Democratization then is more than the legitimization of individual ideas but an understanding that knowledge building requires multiple voices, multiple perspectives, even not-very-well-worked-out perspectives in order to extend understanding and produce new knowledge.

STOP: This article appealed to me because it was written very clearly. Also, I have been considering creating an online forum for grade 6 and 7 students throughout the Surrey SD who attend the Challenge program. The biggest question we are asked is ‘what comes next after Challenge?’ Also, some our low income schools have so many learning needs, are so poorly supported that ‘high’ students are bored, left to their own devices and do not have opportunities to participate with open-ended math with like-minded peers. This could provide continued challenging mathematical material and promote intellectual risk taking. I wondered about posting solutions and role of teacher and so this article showed me that I can teach an eight-week module and that it is possible for students negotiate meanings amongst themselves. The distinction between community of learners and knowledge building particularly interests me. The former seems almost like ‘make-believe play’ where one is ‘acting’ like an expert. For some, this idea is enough to motivate students-they have the buy in but I can see how for other students, acting like an expert does not appeal to them. I think this article also shows me that I would have to be somewhat focused on a topic; for instance, the pre-learning to the online forum was about linear functions-activities were all centered around this. The article included sample activities and I have done some of these. This article prompts me to want to create a proposal for a class--I have to really think about to focus on for grade 6 and 7s. Since many of you have taught this age group what would you focus on? (Geometry? Patterning and Linear Functions since it leads to generalization which is central to mathematics?) Also, Knowledge Forum was text-based but I wonder how students could show learning with multiple representations-for instance, students could draw solutions, post videos etc. I wonder what this would enable? What new problems could arise from multiple representations? – for instance, it could be more difficult to critique multiple representations than just text explanations.

*of course, what about students just googling solutions?

I would love to hear your thoughts!

Multimodality in Mathematics

Article: Ferrara, Francesca. "How Multimodality Works in Mathematical Activity: Young Children Graphing Motion." International Journal of Science & Mathematics Education 12.4 (2014).

This article is based on the theory of embodiment; that we do not just use our brains to think but rather use our whole bodies. Ferrera defines multimodality in two ways. First, from the bio-psychological perspective, there are multi-modal neurons that are active during both seeing and performing an action; furthermore, the same neurons are active when we imagine seeing or performing an action. Second, by looking at interaction and communication in a classroom multimodality is the multiple means we use for making meanings. Mathematical imagination is “entertaining the possibilities for action; entertaining (in the sense of holding or keeping) a state of readiness for enactment of possible actions (Nemirovskey & Ferrara, 2009, p.159 qtd. in Ferrera). It has been established that multimodality plays a role in mathematics learning, in this paper Ferrera focuses on how this takes place. The research question is ‘how does multimodality work in mathematical activity?’ and particularly, ‘what is the role of imagination in multimodality?’

In order to explore this question, microgenetic analysis was used in order to study the genesis of ideas by a subject; in this case, analyze two episodes with one student named Benny. Technology was used in the classroom such that a student could move on one side of the room and a graph (on a Cartesian plane) of their motion would be generated on a screen. Benny’s actions and words were analyzed in order to explore the ways that perceptual, motor and imaginary activities generate ways of understanding and communicating in mathematics. In both cases, Benny was able to connect his movement with the graph modelling it. In the first example, Benny’s thinking process and communication were multimodal as he used words, movement and hand gestures as he developed an explanation about how the graph and his movement were connected. He used the word ‘pretend’ to show an imaginary situation and in doing so, invited others (interlocutor) to engage in his act of imagination. In this way imagination formed a place for inter-subjectivity. Other students who observed Benny move also moved back and forth between the graphical space and physical space of the movement showing that imagining can also help meaning making in mathematics. Also, when Benny wrote about his experience it was as though he was interacting with an ‘imaginary interlocutor’-he was explaining his thinking to a reader.

Multimodality is made of a percepto-motor imaginary that involves a sense of immersion in the experience of doing mathematics in which one forgets there is a passive learner and becomes a unique learner who actively knows, understands and interacts with others in the social classroom. Understanding multimodality has two pedagogical implications. First, it shows that researchers recognize and grasp the complexity and intensity of mathematics learning in the classroom as an inventive act. Secondly, it makes space for creativity in teaching-as imagination is an essential part of multimodality, approaches that provoke experiences with bodily engagement may be effective in mathematics.

STOP:

To me, one very interesting aspect of multimodality was the way that the same neurons (neural pathways) are active when we both perceive (sense) and act (motion) something, and when we imagine sense or acting with that thing. The example cited in this article was picking a rose and smelling it. This runs contrary to mind-body dualism. It also is contrary to the simplistic diagram of the brain cortex where there is a somatosensory region and motor region. The connection between multimodal neurons with multimodality in mathematics learning, made me wonder about the role of psychological research in mathematics education.

My question is how has brain research, or even just new ways of understanding the brain, resulted in (major) shifts in mathematics education research?

Sociomathematical norms

 Sociomathematical Norms, Argumentation, and Autonomy in MathematicsAuthor(s): Erna Yackel and Paul CobbSource: Journal for Research in Mathematics Education, Vol. 27, No. 4 (Jul., 1996), pp. 458-477

Sociomathematical norms are the normative aspect of mathematical discussions that are specific to a students’ mathematical activity. In this paper, Yackel and Cobb use sociomathematical norms in order to account for how students develop a mathematical disposition; that is how students develop specific mathematical beliefs and values and how they become intellectually autonomous in mathematics. The theoretical perspective is based on constructivism, symbolic interactionism and ethnomethodology. According to this framework, learning in mathematics is not about transmitting knowledge but rather about participating in a culture of mathematics. The development of an individual’s reasoning and sense-making cannot be separated from their participation in the interactive making of taken-as-shared mathematical meanings.

The relationship between teachers and students involves reflexivity as students contribute to classroom discussions and teachers legitimize responses. As this happens students’ understandings evolve and teacher’s interpretations of students’ understandings also evolve. For instance, in a grade 2 classroom students are encouraged to solve problems in a ‘different way’ in order to decompose numbers in several ways. This is not explicitly defined for students but as they respond to teacher and listen to others’ responses they learn what constitutes a ‘different way.’ Also, although elegance or sophistication of students’ responses are not explicitly discussed, students and teacher develop ideas about mathematical solutions that are more efficient than others. Also, reflexivity is highlighted as the teacher’s notions of student understandings evolve.

Explanations and actions in mathematics should be based on mathematical thinking rather than status-based; the goal is for students to base their reasoning on mathematical thinking rather than rely on social cues. The goal is intellectual autonomy.

STOP and Question:

This paper talked about sociomathematical norms that characterize an inquiry mathematics classroom. For example, offering a different solution or treating an explanation as an object for discussion. What sociomathematical norms do you think constitute and inquiry mathematics class?  What do you think hinders students from adhering to sociomathematical norms (adopting mathematical ways of knowing)?

Gender and Mathematics-Comparing Sweden and Australia

Source:

Brandell, Gerd, Gilah Leder, and Peter Nyström. "Gender and mathematics: recent development from a Swedish perspective." ZDM 39.3 (2007): 235-250.

Studies in gender and mathematics began in the 1970s as math was recognized as a male-dominated field. Since then, there have been policy and educational initiatives promoting girls to study math. Researchers have also recognized that males and females do not represent homogenous groups; that intersections between gender, class, cultural diversity all play a factor in attitudes towards math and that researchers personal beliefs and theoretical orientations are reflected in their planning, executing and interpreting of research.

A new instrument called ‘Who in Mathematics’ was designed to capture gender stereotyped attitudes among students related to various aspects of math in education and future life. This scale allows math to be viewed as female, male or gender neutral. Results from Sweden were compared to earlier results from Australia.

Overall, girls’ success in math was attributed to hard work and girls were recognized as wanting to understand their work rather than talent. Also, responses demonstrated a conflict between rhetoric and reality. When explicitly asked about gender and math students would respond with equity-based responses but when asked about reality they expressed that math is more male-dominated. One male student expressed how his descriptions of reality were not politically correct. Australian students were more likely to perceive math more as a female domain and more likely to see math as a central to females’ job prospects; this could be attributed to more energetic equity policy. Swedish students were more likely to perceive math as male dominated and researchers link this to how Swedish campaigns are more directed towards science and technology.

The authors conclude by stating that teachers can certainly administer this test as it is fairly straightforward. I think this would be very interesting to do in my work. The enrichment program I work with accepts 2000 students per year and has about 4000 students referred ach year. Other enrichment teachers and myself already collect data and find that more males are recommended to math sessions each year and with every age group (ages 8-12) and at every site (there are 5) in different SES areas; the exception is high SES area where the numbers are more equal. I also notice that in low SES sites, students show less confidence and self-efficacy overall; this correlates with research by Sternberg and Arroyo. I also see girls less likely to offer opinions in math classes. I often wonder about the intersections between SES, gender and cultural diversity. I think it would be very interesting to look into the attitudes of students, especially as the learners I work with show some evidence of strength in math and pursuing a career in math is a conceivable life outcome for them.

Have you seen policy and educational initiatives geared towards math for girls in your teaching/researching contexts? As this article looked at Sweden and Australia, I would be very interested to gain an international perspective on this issue.

Have you seen intriguing observations that could possibly to linked to these initiatives?

Teaching Mathematics for Social Justice

Title: Talking about Teaching Mathematics for Social Justice

Author(s): David Stocker and David Wagner

Source: For the Learning of Mathematics, Vol. 27, No. 3 (Nov., 2007), pp. 17-21

This article is a conversation between David Stocker who (at the time of the article) taught grades 7 & 8 teacher in Toronto and David Wagner, who teaches undergraduate and graduate students and authored a curriculum book called, “Math that Matters” in 2006. For me, the article highlighted some tensions that exist within the area of teaching for social justice and how it relates it with mathematics education.

Tension 1: Ends-based versus process-based guidelines for social justice teaching

First, the authors negotiate the definition of social justice and peace. Both agree that non-violent approaches to conflict and democratic decision making are given. Stocker says that “the elimination of barriers to social, economic, and political inclusion based on race, class, gender, sexuality, ethnicity, religion or ability” is a guiding principle for social justice. However, what is interesting is that Wagner problematizes this by distinguishing between process-based and ends-based visions for social justice teaching. Wagner does not focus on end-goals like ‘elimination of barriers’ because many people throughout history thought they were doing good for others with an end-goal in mind and because this outcome is impossible to achieve. (This reminded me of Canadian residential schools which Aboriginal students were forced to attend.) Rather than focusing on end-goals like elimination of barriers, Wagner chooses to focus on the processes we engage in order to enact our visions for social justice and peace-educating for awareness can be a form of non-violent resistance.

Tension 2: Protecting children versus perspective teaching

One argument against teaching about social justice issues like poverty, war or racial profiling is that we should let kids be kids. Immaculate Namukasa brought this up at the CMESG discussion group; she was born in a country with child soldiers. On the other hand, power-brokers (like those in big business) do not think about others and so we should teach students about different perspectives.  Also, the children we teach are dealing with issues (such as racial profiling, domestic abuse or poverty) and so by not confronting issues we could be disempowering them or silencing them further. Also, teaching social justice does not need to be depressing but rather we can treat students as agents of change and like they matter.

Tension 3: Teachers pointing students’ attention in particular ways is an act of power versus promoting students' personal agency

As teachers, directing students’ attention is an act of power. Providing meaningless contexts for mathematical application while asserting that mathematics is useful for addressing meaningful contexts can be seen as a low level of social abuse. If we work from outcomes based curriculum, this type of low level social abuse is inevitable; but forcing a exclusive social justice agenda or sole pure mathematics agenda is also a problem. As in the words of an Aboriginal elder, we are trying to have our students use their “common sense-that is their sense of the world and their place in it” (qtd. on page 20).

Tension 4: Balance

These authors think that as educators, teaching social justice is our responsibility. They both lean towards problem-based education instead of outcomes-based education. But play is also a part of learning. Sometimes use the word ‘balance’ as an excuse not to reflect on their teaching or as an excuse to water down their social justice teaching. The authors quote Malcom X “An extreme illness cannot be cured with a moderate medicine” (qtd. p. 20).

STOP:

I was really struck by Immaculate Namukasa’s idea in this article; why would we teach our children about horrible situations? It reminded me of a conversation in EDCP 566; we talked about Natasha Levinson and her idea of ‘natality’ and ‘being born into the world.’ Our students are born into the world as it is now, with our current political crises, environmental and social issues. We have to tread lightly in order to make space for hope and in order to make space for students to imagine the world differently.

I have tried to incorporate multiple perspectives social justice in various classes. I have found that when I try to promote my own agenda it doesn’t work but when we have open dialogue and I really let my students take the lead it tends to work a bit better. I have also learned that teaching about social justice requires trust.  I introduced the image on the left in a social justice art class a few weeks ago. We talked about mathematical principles to understand the quote and then I just let my students (all were girls) share their experiences. It was very hard to stop myself from jumping in, defining what the students were talking about and taking the conversation in some tangential direction. I finally concluded the classes’ discussion by saying I have been a girl for a while so I have been thinking and talking about these issues for long time. For an eleven year olds the issues were about a boy blocking them in the hallway all the time (a form of violence), proving themselves to be good at a sport and respected (having a legitimacy to enter a field) and navigating ‘liking’ a boy but being better than them at school (socialization and competition). I felt really good about how our class had developed trust so that they could share and that they told their stories through art. They also had a mathematical statistic spark the discussion.

On the messy side of things, I had one student say her least favorite part of the class was class discussions (but that she liked the making art part)-she didn’t get it. To me this highlighted that avoiding ‘group think’ when teaching about social justice is also important. Promoting critical thinking and personal engagement is the point of education. Overall, I had wonderful feedback from students and parents but that one student makes us think as educators.

I also realized that I have David Wagner’s curriculum book ‘Math that Matters’. I was excited when I first got it (I liked the Noddings and Freire quotes) but I never used it. It made me wonder about if I have an underlying idea that there is a schism between math and social justice?... it took me a while to realize that what we did in art class was related to math. How much math do we have to do in order to make it social justice in math education?  It made me wonder if it is because I have a pre-conceived notion about the types of kids who go math classes? I think addressing how to introverted students fare in social justice teaching would also be an interesting point of departure.

Question:

Is there a particular tension about social justice teaching that resonates with you? Is there an idea from this summary that you have thought about?