On "Sociomathemtical Norms, Argumentation , and Autonomy in Mathematics"

SOCIOMATHEMATICAL NORMS, ARGUMENTATION, AND AUTONOMY IN MATHEMATICS

By Erna Yackel And Paul Cobb

In order to make sense of mathematics learning and teaching, the authors put forward a framework to interpret students’ learning from the perspective of sociomathematical norms. Sociomathematical norms, described by the authors, are unique to mathematics, in contrast with social norms which are general across all subject areas. “For example, the understanding that students are expected to explain their solutions and their ways of thinking is a social norm, whereas the understanding of what counts as an acceptable mathematical explanation is a sociomathematical norm” (p. 461). In this paper, the sociomathematical norms of mathematical difference and mathematical sophistication are established. The central idea of sociomathematical norms is that the normative aspect of mathematics discussion in classrooms is co-constructed by the teacher and students, which opposes the notion that mathematics learning is context-free. The author argues that what becomes mathematically normative in a classroom is shaped by the teaching goals, understanding of constraints, and negotiation among the classroom participants.

Further, the author illustrated how different sociomathematical norms regulate mathematical argumentation and provide extensive learning opportunities for both the students and the teacher. Whiles students consistently contribute to the classroom discussion by explaining their solutions, the teacher is offered opportunities in turn to develop notions of what is sophisticated and efficient for the children. The paper also discussed the link between sociomathematics norms and being autonomous learners. The author explained that autonomy takes place when students take responsibility to an extent beyond “being a student”. Such a transition requires awareness of sociomathematical aspects created by students and teachers.  This is also noticed in many inquiry-based approach classes I have observed.

The authors argued that “initially, students' explanations may have a social rather than a mathematical basis.” (p. 467) While I had a very vague conception of the mentioned process, I experienced an enjoyable enlightenment reading this paragraph. One particular example was used for clarifying how the teacher and students interactively constitute what counts as an acceptable explanation and justification. In example 4, a student changed her answers and challenged the mathematical basis for explanations. The teacher explicitly acknowledged that “she changed her answers on the basis of her interpretation of the social situation rather than on mathematical reasoning, the teacher invents a scenario to clarify his expectations for this class.” (p. 468). I sense that this practice of the teacher was surprisingly effective and powerful since it created a concrete case that every student can comprehend and refer to.

Question:

The author distinguishes sociomathematical norms from social norms. How do you think norms in math classrooms differ from norms in other subjects?

Gender and Mathematics: recent development from a Swedish perspective

Gender and Mathematics: recent development from a Swedish perspective

By Gerd Brandell, Gilah Lede, Peter Nystrom

In the past, research attention to gender equity has been mainly drawn to areas in which females appear disadvantaged and underrepresented. However, during the time of this study, more considerable attention has been placed on educational disadvantages faced by boys. Concerns of boys’ disadvantages, even in the fields of math and science which are traditionally perceived as male-dominated, are receiving increasing attention from the media and researchers in Sweden. Mats Björnsson (2005) noted that focusing on girls’ needs in the classroom could somehow be a reason for boys’ declining results in school. As a result, Björnsson suggested looking into pedagogical reform and other factors that influence education among boys. 747 participants were chosen in the questionnaire study, and 24 among them were selected for interviews. For a majority of the investigated items, such identifying which gender enjoys math more, more than half of the students replied with "no difference" between girls and boys. The study also revealed that female students are perceived as hard-working while their male classmates are associated with disrupting class.

A comparison of this Swedish study with Australian data shows that mathematics is also perceived as a mostly male gendered field at the secondary level. However, Australian students were more inclined to view mathematics as a female gendered domain when compared to Swedish students, and female students also enjoyed math more than Swedish students. The finding seems to agree with the assumption described by Elizabeth Fennema and Julia Sherman (1976): the less a female stereotypes mathematics as a male domain, the more likely she would be to study and learn mathematics.

It is quite interesting to notice that female students from different countries perceive math differently. The study did not investigate very deeply into this complexity. I have some teaching experience related to this. For years, I have heard stories that female students struggle more with high school math than boys in China. It is a common impression that math is a male-gendered subject. Interestingly, several students who hated and were scared of math in China found math quite enjoyable when learning in countries as the US and Canada. I would assume that it might be attributed to the less challenging curriculum used in newer schools in the West. This might be true for elementary level math, but I don’t see a huge difference in difficulty and depth between countries when comparing grade 11 text books.

Do you have similar teaching experiences, and what do you think?

On "Teaching Mathematics for Social Justice"

Wagner and Stocker trace their interests in social justice back to their family upbringings and life experiences. Stocker was raised up in an Adlerian family in which punishment and reward are replaced by democratic approaches. On the other hand, Wagner was raised up in a Mennonite family (a Christian denomination) but changed drastically after backpacking around the world and saw disparities and injustices. In the conversation, Wagner and Stocker exchanged ideas over what math educators must do with social justice.

Principles of peace and social justice

Stocker describes 3 basic principles regarding peace and social justice:

- non-violent approaches to conflict

- democratic decision-making processes

- the elimination of barriers to social, economic, and political inclusion based on race, class, gender, ethnicity, religion or ability

Wagner negotiates the definition of social justice by pointing that much violence is wrought by people with good intentions, and therefore the elimination of structural barriers as an “ends-based” vision should be practiced with caution as “others would be involved in when addressing violence”. He prefers to focus on processes rather than end goals when integrating social justice in teaching mathematics. Stocker refutes this sentiment by noting that it is “extremely competent” businessmen who are maximizing profit for shareholders without sparing thought for “good intentions”.

Exposing children to the real world

In arguments over whether children should be shielded from exposure to social justice and inequities such as race and gender, Stocker claims that children are potential future makers for the world and it is the teachers’ responsibility to direct the eyes of children to these real world issues. Wagner agrees that children should see the negative and broken aspects of the world but he questions the intention of dragging children into the battle of social injustice. Further, Wagner concerns that it is a subtle form of “social abuse” for teachers to force their political and social agendas onto children. 

Balance of teaching

Wagner discusses the need for balance between teaching actual mathematics and the social justice component to mathematics. Stocker notes that there is no semblance of balance in current curricula, comparing the “balance” to the “balance” between an elephant and a pea. In reality, Stocker is not concerned about the balance, but rather to the many teachers who use the concept of “balance” as an excuse to avoid bringing social justice teaching to the classroom. Wagner and Stocker both agree that perspective teaching must also be included to bring balance to teaching mathematics, and emphasized the importance of the social function of play in learning.

Stop:

It has been a struggle for me to balance these elements of mathematics education in the classroom. On one hand, I want my students to know what their ideal world should be by downplaying the severity of issues around us. On the other hand, I know that some issues such as pollution, poverty, and inequity in education itself will only get worse by ignoring them. According to Wagner, the world is always changing, and a “better world” is an end goal in itself, meaning that people working with good intentions towards this end goal have the ability to cause more damage than they fix. I share the belief that implies that each generation has their own contributions and responsibilities; adults who are interested in social justice should take responsibility to “fix it”. 

When enacting social justice in classrooms, how do you address the question and justify of whose responsibility is it (everyone’s or adults’) to fix these issues and make the world a better place?

On " The Linguistic Challenges of Mathematics Teaching and Learning: A research Review" by Mary Schleppegrell (2007)

The article synthesizes research from the mid-1980s to 2005 in the fields of linguistics and mathematics education.  The article highlights several linguistic challenges of math learning:

1.       Multi-semioticity: the construction of the meaning of math concepts is drawn from multiple sources including symbols, oral language, written language and visual representation, order, position, etc.

2.       Specific grammar features in math language: One feature is that the math language uses noun-dense phrases such as “the volume of a rectangular prism with sides 8, 10, and 12cm”.

3.       “Mathematics is highly technical, with characteristic patterns of vocabulary and grammar” (p. 142). An example was used to illustrate the idea. “George has twice as less money than Tina” is considered technically incorrect, in comparison with a newspaper report “traffic in Sydney during peak hours is nine times slower than in Melbourne”.

4.       Precision of mathematical conjunctions to link logical elements: words such as if, when, then are used in precise ways in developing theorems and proofs and they represent different ideas compared to use in everyday life such as in “if I got an A, I would be very happy”.    

I appreciate the notion that a key challenge in math teaching is to help students move form informal discourse to formal discourse which is necessary for disciplinary and interdisciplinary learning in school. Especially with the inquiry-based approach to encourage students to connect math concepts with everyday life, students are comfortable bringing their “everyday” language to participate in the activities. In addition to teaching formal technical terms and demonstrating the language used in mathematical discussion, the article suggests to explicitly teach the language and bring self-regulated awareness of “formal” math talk in classrooms. It is a very insightful article as it casts doubt on the common understanding of the independence of effective math learning over mathematics language.      

Question:

As the author indicated, most teachers have recognized that technical vocabulary is a challenge, but many do not pay enough attention to the grammatical patterns. How should we offer help in the classroom to ELL learners who are new to both “everyday” English and “mathematics” English?

On "A Linguistic and Narrative View of Word Problems in Mathematics Education"

Gerofsky explores the goals of using world problems in school by using linguistic analysis on the structure of mathematics word problems used in schools. There are usually 3 components seen in a typical word problem: 1. the exposition, 2. some information to solve the problem, and 3. the question. Gerofsky pays special attention to the first component which establishes the characters and places of the story, and argued that it only worked as a statement which is actually irrelevant to the question. The author continued to examine the structure of word problems with the lens of linguistics analysis. It is pointed out that there is a mutual understating over the inconsistency of the tense used in word questions (sometimes, it is all present tense, while in others it is mixed with present and future tense), which might be explained by the philosophy behind word problems: hypothetical solutions to hypothetical situations. Gerofsky argued that most questions could be rewritten in a standardized form: Suppose condition A exists. Then, if conditions B and C held, what would be the answer to D. This leads to an interesting argument that word problems are hardly a fiction genre as word problems don’t present “truth values”. Gerofsky invites us to think of word problems as parables, but also cited contradictory studies about the analogy. According to her, delineating the boundaries of word problems as a genre opens a door to generate more productive discussion over the rationale for using word problems at school.

Stop:

While many teachers hold the belief that word problems help students to learn “adaptive” or “contextual” mathematics, the author invites us to look closely at the elements, structure and variance of world problems. There is no doubt that students have developed an “immune system” to word questions, as they skip the story and quickly search for numbers and key phrases which are useful for plugging in to formulas. With that mindset, students gain a limited understanding of goal setting, planning, and do not develop a positive attitude toward problems.

Question:

Do you prefer “real world” word problems with the three elements mentioned, or abstract problems that cut to the chase?

My Research Interest

I am interested in how technologies have helped students’ problem solving. I have been actively involved in developing and teaching an extra-curricular robotics and coding program, offered in libraries and learning centers. Although I observed a proportion of students in these programs demonstrating significant improvements in engagement, more research is needed to investigate the contributions of technologies to mathematical problem-solving. 

On "Mathematics as Medicine" & "Balancing Equations and Culture"

Article A: Doolittle (Canada): Mathematics as Medicine (in Proceedings of the Canadian Mathematics Education Study Group (CMESG) 2006, pp. 17-25).

Edward Doolittle from the University of Regina is a mathematician and a Mohawk Indian who grew up in the suburbs of Hamilton, Ontario, knowing almost nothing of his indigenous culture before attending the University of Toronto. He was connected with indigenous traditions upon participating in an Indian Health Careers Program, a program designed to help to increase the representation of Aboriginal people in medicine and other health-related careers. From there, he started to seek efficient research methods to help mathematics education for indigenous people.

He is skeptical that quantitative research efforts, such as quantifying or justifying assessments with standardized outcome, can have fruitful results. Doolittle argued that these approaches would not be able to offer adequate solutions as the complexity of the situation seems to be beyond the capacity of any research which tends to offer simple responses to a complex question.

Ethnomathematics is a mean of addressing the interactions between math thinking and cultural beliefs, and the presenter found it far more reflective and respectful to Indigenous traditions of thought. However, he reminded researchers to be aware of the tendency of oversimplifying the inquiry such as saying “The tipi is a cone”. Another two pieces of advice include using culture-appropriate language and words that blend in with the indigenous language when introducing mathematics terms, and respecting indigenous spiritual traditions by trying to view mathematics as a power of medicine from the perspective of it connecting lives and making us better people.

Article B: b) Doolittle & Glanfield (2007): Balancing Equations and Culture: Indigenous Educators Reflect on Mathematics Education. FLM, pp. 27-30.

In this conversation between Doolittle and Florence Glanfield, two aboriginal mathematics educators, several issues relating to perspectives on the value of mathematics were discussed.

1. Doolittle elaborated on a story in which a boy came back to his camp with a strange animal, a horse. The boy showed his people how to make use of and keep a good relationship with the horse. The horse became an asset to the camp and the boy later became the chief. Doolittle raised the question of whether math can be as useful as the horse, or would it rather be a Trojan horse which might actually cause huge loss to the camp.
2. Florence was concerned about the power of mathematics which she felt that she benefitted from, but was not able to be distributed equally to her tribe. She recognized that there was a division and barrier between “the secret mathematics society” and the mass public.
3. Doolittle asked the question of whether mathematics is universally powerful or only has power in western contexts. He was skeptical that many requirements for math in the job market were just arbitrary and did not serve a real purpose.
4. Florence believed many pre-service teachers were not offered opportunities to explore relationships within math. They were “silenced” in their early formal years in expressing their ideas about math notions and relationships.
5. Doolittle found that the aboriginal perception of relationships was not compatible with mainstream relationships presented in math. He was concerned that current prevailing school mathematics might force aboriginal people to give up their natural and innate mathematical understanding of the world.

They both believed a balanced approach towards math from the lens of aboriginal culture. Mind, body, emotion and spirituality can be most promising to help both aboriginal people and math society.

Reflection: :

A sample used to illustrate the complexity of mathematics in indigenous society  is the following,

Q: If he gets four dollars a day, how many is he going to have in two days?

A: Six.

Although the answer appears wrong at first glance, there are more subtle ethnic implications behind the answer. For example, if a labourer were to work for two days and was told that they were to receive 4 dollars each day, but only ended up receiving 6 dollars in total, this question would be treated as “real math” and the answer would end up being a reflection on reality. One reason that “street math” is not a major chapter of our current curricula is that “street math” is more of a personal or group interpretation of a phenomenon with minimum abstraction and lack of ability of generalization. Of course this is against some of the perception of math and its value. Since we have limited the term of math to the western context, I kind of feel some new names should be introduced to recognize the different “original math” base on its culture roots.   

Question: 

In teaching current school math curricula, have you experienced any hesitation, resistance or conflict due to cultural differencecs?