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?

Linguistic Challenges in Mathematics

Citation: Mary J Schleppegrell (2007): The Linguistic Challenges of Mathematics Teaching and Learning: A Research Review, Reading & Writing Quarterly, 23:2, 139-159

The study of linguistics and mathematics education is based on the premise that we use language in order to participate in ways of knowing. Language and learning cannot be separated. In the case of mathematics, the goal of the teacher is to move students from more informal (everyday) ways of using language and understanding to more formal ways of understanding. In the case of mathematics education, this means the goal is for students to use more technical language of the discipline in effective ways; this formal language is called the ‘mathematical register’.

A mathematical register, “is a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings. We can refer to a ‘mathematics register’, in the sense of the meanings that belong to the language of mathematics (the mathematical use of natural language, that is: not mathematics itself), and that a language must express if it is being used for mathematical purposes. (p. 195)

The mathematical register includes semiotic systems such as symbols, oral language, written language graphs and diagrams. It also includes grammatical patterns. One central aspect of Schleppegrell’s research review is that language issues is math are not solely about terminology or specialized terminology. Rather mathematical reasoning uses grammatical patterns that are dense and quite different from informal language use. Mathematical thinking is not just about using precise terms but the way that one uses terms must also be precise. For example, in explanations, teachers or students might use number operations (such as squaring) as nouns rather than verbs. Teachers may also fail to recognize that students use terminology in ways that are different from the ways that teachers intend it to be used. Students tended to use mathematical terminology as description, sequence or choice while teachers use mathematical terminology for classification, principles and evaluation. In one example a teacher and student pair are working with tangrams. The teacher uses the term parallelogram as a classification but the student uses it as a description.

Some strategies for developing the mathematical register include:

-interactive activities that allow students to construct meaningful discourse about mathematics

-verbal explanation of word problems

-develop connections between everyday meanings of words and their mathematical meanings

-recognize and use technical language rather than informal language wen defining and explaining concepts

-explicitly evaluate the use of technical language

-use writing but be explicit about not writing in a narrative form. Rather use other forms of writing such as procedures, descriptions/classifications, explanations/findings and arguments about math theorems

There are two tensions that teachers encounter as they teach the math. One is the dilemma of mediation: this is when a teacher has to decide when it is best to validate the perspective of the learner by listening and when to introduce more formal ways of explaining concepts. The second is the dilemma of transparency: this is when a teacher has to decide when to talk and when not to talk. Scleppenberg also described how focus on linguistics is important for low SES and culturally diverse learners.

STOP:

One idea that I took away from this article is about the effective use of writing in mathematics teaching. A few weeks ago, in my post about ethno-mathematics, I questioned if the discussions and writing I am using in math class is working for ELL learners. Last week, the Marion and Walter article talked about creating a culture of mathematicians by having students create math academic journals that are posted up in the room and critiqued by peers. Schleppegrell’s article states that in order to use writing to further mathematical understanding it should not just be narrative but rather teacher should explicitly teach students how to use writing for procedural and explanatory purposes. To me, this reminded me of our conversation about genre last week and how important it is to teach different genres.

It also made me think about primary learners and how sometimes their math journal writing seems to show pseudo-mathematical understanding. This article makes me wonder if this is because young students are writing in a narrative way? Developing a sense of story is a huge component of Kindergarten curriculum.  Furthermore, students only recently learned that the orientation of a symbol changes its meaning: ‘p’ is different from ‘q’ is different from ‘b’ is different than ‘d’. Now, we expect them to understand that numbers don’t magically change but that we perform operations on them and then they change.  That a 6 and 7 are nouns, but that a + and – are verbs. Looking at this developmentally, I can see how this would be extremely confusing to a young person.

Question:

There is a common held assumption that math as the least language-dependent subject. What do you think? Agree or disagree? How did this assumption come to be?