Beliefs and Norms in the Mathematics Classroom

Author: Yackel, E., & Rasmussen, C.

Source: Yackel, E., & Rasmussen, C. (2002). Beliefs and norms in the mathematics classroom. In Beliefs: A hidden variable in mathematics education? (pp. 313-330). Springer Netherlands.

In this chapter, the authors investigate how students’ mathematical beliefs and sociomathematical norms are changed in mathematics classrooms from both sociological and psychological perspectives. They analyze the comments from the students that are reflected on their mathematics activities, and the dialogues among the instructor and the undergraduate students in differential equations classes that the instructor use an inquiry approach.

According to the authors, what becomes normative relies on what accepts mathematical explanations in classrooms. They describe two aspects of acceptability of explanations. The first is that an explanation is acceptable when it provides a clarifying function that makes clear other one’s thinking as one of communications. The other one is that an explanation should be “about student’ mathematical activity with entities that are part of their mathematical world.” (p.325) One of examples the authors show in this article is that the participated students dynamically recognized the criteria for the acceptable explanations and adapted their answers when the instructor shifted from a calculation orientation to conceptual orientation in his questions in the differential equations class. In short, through the observations, the authors conclude social and sociomathematical norms and individual beliefs have strong relationships and evolve together in classrooms.

 I agree with the idea that the beliefs students hold and (social and) sociomathematical norms are developing each other. Students might constitute own beliefs of mathematics while socialmathematical norms change in classroom. But something that was not included in this article, actually my curious point, is whether teachers/instructors change their beliefs or not in classrooms. Do they fix their beliefs? Or do they modify/adapt their beliefs with the development of socialmathematical norms? I believe they also do so but they do not often change their belief regard to the dialogues in the classroom.

 Additionally, I think only teachers lead to develop sociomathematical norms. Although the authors explain students also constitute the sociomathematical norms in their interactions, no change might be occurred in the norms if teacher is not in the classroom. This implies the current sociomathematical norms are students-centered. Hence, my another question is, can students mainly lead to develop sociomathematical norms with minimal supports from teachers?

  Finally, because I am studying immigrants education, I am curious how students who study math in their additional language interpret sociomathematical norms. Some of them may not understand native students’ conversations which constitute sociomathematical norms. In this case, I expect they recognize what “bad” explanation is rather than explore “good” explanation from nonverbal communication. But I do not think they can interpret it as well as native students.

Question:

What do you think about the classroom that students proactively lead to develop sociolmathematical norms with just minimal observations and supports from a teacher? Do you think it is dangerous? Or effective to their autonomy, creativity etc.?

Difference, Cognition, and Mathematics Education.

Author: Valerie Walkerdine

Source: For the learning of mathematics, 10 (3), 51-56.

As we discussed many times in this course, this article starts with the controversial idea that the mathematics children do in classrooms is higher level than the mathematics street children do to sell the merchandises. The author, Valerie Walkerdine, questions what the higher level is and how we can make sense this nature of mathematics and mathematics education. She argues that school mathematics has nature “to regulate and control through reason in a social order” (p.54) that colonizers hold and they do not have to calculate to survive. However, she addresses the essential idea of mathematics as reason became sanctuarized within the curriculum now, which leads to that everything becomes mathematics. About the sequence which takes us pre-logical to logico-mathematical reasoning, she describes it as “a discursive relation in a new set of practices … with its own modes of regulation and subjectification” (p.54) rather than from concrete to abstract.

   In the last parts of this article, the author mainly points the pain that female students feel when moving from one mathematical practice to another. One example the author shows is that the girls who scored high IQ and were positioned as “clever” at 4-year-old by the mother, became to be positioned as “stupid” by the teacher at 10-year-old in the school. To survive from this unendurable pain, they recognize themselves as the target of violence of the others (especially boys). The author argues that even though girls displayed the remarkable characteristics, it does not mean they are in success.

In case of Japan, I suppose Japanese children get accustomed with school mathematics in early stage. Japanese mathematics classes devote a higher proportion of the time on abstract problems compared other countries, and some children start to go or take math tutors (such as abacus) whey they are in pre-school. Although I have to investigate how exactly they are, I wonder if there is a reverse transition of Walkerdine’s idea in moving from one practice to another, which means from abstract to concrete.

Additionally, I cannot agree with the author’s argument that high-performance girls are designated as only hard-working whereas poor performance boys are designated as bright even they do not show the evidence. It has been over 20 years since this article published, it might not be true in the current society. As far as I know, it is definitely not true at least in Japan, and there are no discriminations of evaluation about academic attainment among gender. However, most female students in Japan do not choose mathematics or science as their main discipline in post-secondary education. I am curious this fact is coming from culturally or biologically.

Questions:

Do you agree with the author’s argument that “high-performing girls came to be designated as ‘only hard-working’ when poorly-achieving boys could be understood as ‘bright’ even though they presented little evidence of high attainment” (p.55)?

Do you believe there are cultural or biological differences about mathematics ability/learning/preference (like or not like math) between sex/gender?

"The Real World As We Have Seen It": Latino/a Parents' Voices on Teaching Mathematics for Social Justice.

Author: Eric Gutstein

Source: Mathematical Thinking and Learning , 8.3 (2006):331-358.

  Gutstein, the author, explores and clarifies the opinions toward to mathematics programs for social justice from the view of Latino/a parents. He implemented four “real world” projects which invite the students to use mathematics and think/analyze the real problems in the world such as product cost of weapons, poverty and wealth, and discrimination of mortgage. His school district holds the high proportion of Latino/s and most of the parents are low income and their mother tongue is Spanish. In this article, he conducted the interviews with 10 Latino/s parents and summarized their voices.

  In short, the Latino/a parents want their children to understand, through education (including mathematics), about the discrimination, injustice and disadvantage which they faced and experienced in the society. Furthermore, the parents believe their children need to know that people who belong to the marginalized community have to defend them by themselves in the above society. In terms of mathematics, the parents see “mathematics as an integrated part of life” (p.352) and have a flexible idea to understand the role of mathematics in life such as different interest rates among races. The purpose of education for the Latino/a students, their parents think, should be preparation for the above injustice world while the parents agree with the conventional purpose of education. Finally, the parents advocate the educational programs that the author provided to their children, and they do not believe those programs are propagandized.

  As I see the voices of the Latino/a parents in this article, their voices seem to be premised on their marginalization. They emphasis on the way how marginalized community members survive in the suppressed society rather they try to solve this inequity circumstance. I suppose, to reduce this problem, the majority groups should learn social justice from the perspective of the minority groups to change the unbalanced society.

  Although the voices involve little mathematics contents in this article, his programs are very interesting to me. These programs enable all students, even who do not like mathematics or are not good at mathematics, to participate the mathematical activities. Therefore, they can feel mathematics as a part of life, can recover their confidence toward to mathematics, and can find the own motivation to study mathematics. As I mentioned above, participation of dominant groups is necessary to reduce the problem of inequity, thus the participants in those programs should be mixture of all groups. In this case, I am curious about the responses to the program for social justice that both majority groups and minority groups have, and balance between their power: majority vs minority.

Question:

Considering several reasons: increasing teachers’ burden, national curriculum, limitation of time etc., do you think it is possible to involve social justice contents in regular mathematics class realistically?  How can we include them in mathematics class?

Hedges in Mathematics Talk: Linguistic Pointers to Uncertainty

Author: Tim Rowland

Source: Educational Studies in Mathematics, 29(4), 327-353, 1995.

  The author, Tim Rowland, conducted the interviews with 10-12 years old children to linguistically analyze their uncertainty (hedges) in their predictions, generalizations, and explanations within the mathematical discourses. He utilizes the hedge model of Prince et al. (See the image below), and describes several cases of the hedges by both the students and the teacher (the author) that were extracted from the interviews.

(Rowland,1995, p.337)

 

• Ex)
• Plausibility Shield: I think…, maybe, probably
• Attribution Shield: According to N…
• Rounders: about, around, approximately
• Adaptor: a little bit, somewhat, fairly

  In the interview, the participated children often use Rounders and Plausibility Shields to express their proposition and to move their understanding from uncertainty to certainty. He addresses many children use the hedges to protect from being “wrong” since school-culture leads students to believe mathematics is evaluated by only binary: right or wrong. However, he suggests that using of the hedges in students’ conversations can inform their anxiety, fear, or lack of confidence for their understanding. Therefore, it makes teacher to be able to support their development from vagueness to conviction.

  I agree with the idea that school-culture forces students to provide high quality (accurate) opinions and instills fear in them to be “wrong” in mathematics classes, and agree with that the words and phrases of the hedges could help teachers to know their students’ understandings toward to mathematics contents. It might be difficult for teachers to know whether their students certainly understand what they have learned by tests which only have very small spaces to fill students’ explanations, and to know what they are exactly thinking. The interactions in the classroom, regardless of among students or between teacher and student, hold a lot of information about students’ attainments and idea, thus teachers need to be sensitive for their conversations in addition to their answers.

  In this article, Rowland mentions the zone between students’ proposition and conviction. If I could talk to him, I would ask him if there are any zones between nothing and proposition. After they confront prediction, generalization, and explanation in mathematics class, I believe, students might have something or nothing in their mind before they reach to uncertainly phase. In this case, I wonder if they use the hedges as well or use other words and phrases in their conversations.

  Something that was not included in this paper but actually I am interested, is about students who learn mathematics in additional language. Because of their language limitation, their usage of the hedges would be different from other students and other findings might be observed.

Questions:

When you hear prediction/explanation/generalization/ from your students in mathematical discourse, what point do you watch? i.e. facial expressions, voices, other student’s reactions etc.