Presentation: Immigrant Students and Sociomathematical Norms

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Algebraic Thinking in the Early Grades: What Is It?

Author: Carolyn Kieran.

Source: Kieran, C. (2004). Algebraic thinking in the early grades: What is it. The Mathematics Educator, 8(1), 139-151.

The author, Carolyn Kieran, firstly shows the difference between arithmetic thinking and algebraic thinking. She points out students focus on calculating rather than a representation of relations in the shift from arithmetic to algebra. For example, students interpret an equal sign as “a separator between the problem and the solution,” or “a left-to-right directional signal” when computing. (p. 140) She suggests the adjustment is required to develop the way of thinking in algebra with focusing on relations, both representing and solving a problem, doing/undoing, both number and letter, meaning of the equal sign etc.

The author invites us to think about the difference and commonalities of the descriptions of algebraic thinking among several curriculums. While she acknowledges the differences of them, she claims two similarities: an emphasis on “the importance of relationship between quantities”, and a stress on “generalization, justification, problem solving, modeling, and noticing structure.” (p.147)

According to the author, algebraic activities can be categorized into three types: generational, transformational, and global meta-level. Generational activities are making the expressions and equations which contain much of the meaning-building for the objects of algebra. Transformational activities are manipulating activities with keeping equivalence such as operations. Global meta-level activities are “for which algebra is used as a tool but which are not exclusive to algebra” (p.142) such as problem solving, and involve general mathematical processes and activities. These are important especially for the meaning-building in generational activities. The author addresses that the idea that the development of algebraic thinking at the elementary level is the development of these mathematical way of thinking in Korean curriculum, is crucial for the above algebraic activities. She finally claims the global meta-level activities of algebra which involve both for meaning-building and developing-way-of-thinking, can lead mathematics educators to have a vision of algebraic thinking in the elementary level that guides adaptably their students to algebraic activities at the later grades.

I often hear the obstacles that students experience when the transition from arithmetic to algebra, and have experienced my students facing the similar problem when they go to a middle school from an elementary school. One of my students who was not good at mathematics was struggling to cope with equation. He often wrote (added) the operation result of the left side on the right side in equation, and kept trying to solve the equation problem. Therefore, the meaning of an equal sign for him was obviously the same one as the above example in this paper.

The Japanese curriculum does not categorize mathematics into arithmetic, algebra, geometry, and statistics. In fact, I have never seriously thought what algebra is before. Additionally, there are no explicit descriptions about algebra in the mathematics curriculum in Japan while it contains many contents of algebra. I do not think Japanese mathematics teachers exactly understand what algebra is and its features. However, interestingly, Japanese students tend to get higher scores in mathematics in several international academic performance surveys. Thus, my personal questions to myself is whether Japanese mathematics teachers need to understand what algebra and its feature is, to improve their approach (this is not a question for you unless you want to study math education in Japan!).

Questions:

I think there might be other obstacles to transitions in mathematics when students go on to a middle school from an elementary school, to a high school from a middle school, and to a university from a high school. Have you ever faced any other difficulties for your students (or yourself) during the above periods?

Prototypes, Metaphors, Metonymies and Imaginative Rationality in High School Mathematics

Author: Presmeg, N. C.

Source: Presmeg, N. C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23(6), 595-610.

In this article, Presemeg firstly invites the readers some examples of prototypical mathematical images that can be necessary for reasoning in solving mathematical problems in high school classroom. Her study shows the prototypical images acted as metaphors which lead the students to the mathematical reasoning process, and there were many images that intend to metonymic purpose in the students’ thought process. The students used “concrete” images to solve the mathematical problems. For instance, one student idiosyncratically thought of a ship on a water level as the metaphor to remind her to utilize 180 and 360 degrees to solve the acute angles in the trigonometry problem. The author acknowledges the difficulties in the problems of generality that can be observed in metonymies, although she points out the strengths of visual processing (as following).

  The author describes the category of imagery that pure relationships are depicted in a visuospatial scheme was designated as pattern imagery. She explains imagery can be used to illustrate abstract situations by concretizing the referent or by using pattern imagery. Additionally, “the mediating link between specific concrete cases and generalized mathematical concept lies in pattern imagery which shares some of the imagistic nature of concrete images but goes beyond them to distinguish regularities and commonalities and to abstract these in a different kind of image which provides a basis for the formations of the concept.” (p.605) In other words, pattern imagery gives us the generalization of concrete images and invites us different concepts, but they are based on individual leaners’ mathematical experience which can be shared with others.

This article made me rediscover the multiple meanings of image in mathematics. When I introduce the new concepts or contents in the classroom, I usually bring or explain the objects that the student might hold as a commonality. However, through the lessons, the students might channel the mathematical concepts into the image that they are (more) familiar with. Although it is difficult to estimate the all images each student has, I wonder the introduction with the image that the students hold can help the students to understand the mathematical concepts earlier than using other images. Also, how can mathematics teachers utilizes the different types of imagery in mathematical reasoning?

  The other point this article reminds me is the power of images in mathematics classroom. When I explained the meaning of surface area of solid figure to my student who is a Japanese language learner, I firstly use the way of expression that my other Japanese students could understand. (Think about the situation that you soak this solid figure in a fish tank with filled water, the wet areas equal surface areas of this solid) He understood what I said in Japanese, but he still could not understand well what surface area is. After that, I considered more universal objects in the world as much as I can think up and explained him the same situation with a glass of water and a dice, and it finally worked for him. Although this article mentions the images which are embedded in the students’ mind, the images (including metaphors etc.) in math textbooks might be common only for certain groups such as majorities.

Questions:

Do you think the description without images in solving mathematical problem can develop students’ mathematics ability? (For instance, the questions ask some angles in geometry but no images attached)

What do you think about the images in mathematics textbooks? Do you think those images help students to understand mathematics?

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?