The missing paradigm-Students' perceived mathematical norms

In Students' perceived sociomathematical norms: The missing paradigm, Levenson, Tirosh and Tsamir conducted research in 2 Grade 5 classrooms in Israel, looking at the effect of 3 aspects of sociomathematical norms: teachers' endorsed norms, teachers' and students' enacted norms and students' perceived norms.

Both of the classrooms were taught by experienced teachers, one, Rose, had 31 years teaching experience and the other, Hailey, had 14 years. The researchers investigated the sociomathematical norms that were related to mathematically-based (MB) and practically-based (PB) explanations. MB explanations would be based on previously learned material or mathematical properties, whereas PB would include context examples and/or manipulatives.

In Rose's interview, she felt that personally, MB explanations were the most convincing. However, in her classroom, her endorsed norms were to use either a PB explanation or a combination of the two. She explained that by doing so, she was teaching a method the majority would understand. Rose and her students' enacted norms showed she often asked her class to participate, but usually only the high-achieving students would. Students did not comment on each others solutions to problems, nor did they have to comment at all if they chose not to. Also, students only gave MB explanations to Rose's questions, and she would respond by inserting a PB explanation. The students' perceived norms were that Rose used whatever explanation was best for the student, but that the PB explanation was perceived to be for the low-ability students, as it was easier to understand. They did all agree that they could use either explanation and that Rose would accept either as correct.

In Hailey's interview, she also preferred the MB explanation herself. She also felt that the MB explanation was the best one for her classroom and couldn't understand why the PB explanations would be helpful, referring to them several times as "cute". Hailey's endorsed norms support this opinion, as she says she would only use manipulatives to introduce a topic and then would quickly move on to MB explanations. She believes the MB explanations lead to higher-order thinking skills. Hailey believes MB explanations "are more mathematical" (p.181). She feels her job is to

challenge her students, by asking for MB explanations. Hailey's enacted norms showed that many students of varying mathematical ability participated in class, she reminded her students regularly of mathematical properties, and encourages them to solve problems in different ways, although using MB explanations. Students' perceived norms are that Hailey would only ever give MB explanations

to any student, regardless of ability. One student related PB explanations to students of low ability, as some in Rose's class. They did agree that Hailey would accept any type of explanation.

This study really bothered me. I have a lot of differences of opinion with both the teachers, but for very different reasons. Rose, at first reading, seems to be the teacher that is using a variety of strategies in her classroom and is doing her best to reach all her students. However, the way in which she uses the PB explanations in her classroom leads them to be associated with “low” students. She also has a classroom where only the high-achieving students participate. I teach Grade 6 and 7 and am constantly using PB explanations for ALL my students. I find that having a good grasp in pictorial representations of problems means that students can make the leap to abstraction, particularly algebra, much more easily. Indeed, I have given my class problems that are almost impossible to solve without pictorial representations or manipulatives! Also, Rose does not foster participation. Participation is not an assessed part of my classroom, but engagement certainly is. Students should not be punished (in marks) due to their introversion, nor should they be allowed not to engage in the activities. Giving students a chance to work on problems in groups, to explain their thinking in pairs, is a way around the “problem” of participation in whole-class learning.

I also think Hailey is doing a huge disservice to her students. Shunning PB explanations is robbing students of a useful thinking and problem solving tool. MB explanations are not always the only way to challenge your students. Again, there are numerous way to pose very challenging PB questions to encourage PB explanations. I certainly have a problem with PB explanations being called “cute”. Shouldn’t mathematics teachers be teaching a variety of tools so that different types of learners can choose the tool that is best for their style of learning? Certainly some students prefer MB explanations, but by ignoring PB explanations, you are not giving your students a “complete” mathematics toolkit.

The other part that worried me was the students’ perceived norms. Although in both the classrooms, students claimed either MB or PB explanations were acceptable, no student in either class ever offered a PB explanation for any of the questions posed by the teachers. This seems like quite a contradiction. This also makes me wonder if their perception may be that although the teachers would accept the PB explanation, they would prefer and value a MB explanation more.

Hailey strongly believes that MB explanations are best in grades past Grade 2. Do you agree? Do you think MB and PB explanations are mutually exclusive?

Our Structures

Genderism and Math

In Snips and Snails and Puppy Dogs’ Tails: Genderism and Mathematics Education, Indigo Esmonde argues that although arguably the achievement gap between boys and girls in mathematics has been shrunk, and in some cases virtually eliminated, those gaps are being measured using binary gender terms. Binary gender refers to boys and girls and students being classified as one or the other based on biology.

Esmonde points to a recent shift in the Toronto School District specifically, to focus attention back on boys. For many years, a large amount of mathematics research has been focused on girls and narrowing the gap. Now that many goals have arguably been reached, educators are realizing that boys and their learning and achievement has been largely ignored. Boys are now over-represented in learning support, underachieving in class and are exhibiting disruptive and violent behaviours.

Esmonde then goes on to discuss that “sex” and “gender” are terms that seem to be used in research interchangeably and should not be. Indeed, “sex” is determined using biological means, whereas “gender” is a social construct. There are also many students who do not fit into those categories. Past research did not consider many other factors, such as social class, culture, race, and ethnicity. It also often assumed that all girls and boys are virtually identical to each other: girls are emotional and caring, boys like hands-on activities and movement.

Esmonde feels strongly that educators need to work towards an anti-genderist mathematics classroom. This term, in her opinion, does not mean to eliminate gender in classrooms, but rather “the goal should be to challenge the gender binary and pervasive gender-normativity in education.” (p. 30)

Stops:

I found this article extremely interesting and also challenging to my pre-conceived ideas on gender in the classroom. All of my reading on gender education thus far has been focused (unintentionally) on the gender binary, boys vs. girls. In my teaching experience, there does still seem to be a gap between girls’ attitudes and achievement versus boys’ attitudes and achievement, but the gap is certainly much smaller than it was even 10 years ago.

This article has made me reevaluate both my classroom and the gender norms that are reinforced, as well as the resources my school uses in the mathematics classroom. Word problems are often the easiest way to observe the gender-normalizing classroom we create. The texts that I have used, which are strong in terms of content, are full of stereotypes and certainly conform to gender-normative behaviour: boys have short hair and play sports, girls have long hair and cook and bake.

Both this article and last week’s articles are closely connected, as teaching students about social justice includes issues of gender. Moving forward, I would be very interested in reading about and participating in activities that help students understand and appreciate that gender is not simply boys and girls, that it includes many marginalized groups and can be fluid in a person’s life. School is a place where students need to learn about different perspectives and experiences and be taught acceptance of all the differences in gender, race, religion, culture, ethnicity and how to work together to form a community. Educators need to take the time to consider and appreciate their students and the realities they live in and with and show students that every difference is valued.

Do you feel it is necessary to challenge gender-normative behaviour in the mathematics classroom? Do schools have a responsibility to challenge gender norms? How do you see yourself challenging the gender binary?

The Sociopolitical and Math Education

In the article titled The Sociopolitical Turn in Mathematics Education, Rochelle Gutiérrez argues that in recent years, many organizations and researchers have begun to carefully consider and critique the way in which mathematics classrooms and research have tried to work together to build equality, without considering the framing of such efforts, including cultures, ethnicities, gender and background both of the students and of the society creating the math in the first place. Overall, the article wants to highlight the sociopolitical turns and how they could be helpful as well as hinder.

She mentions several theories, many of which centre upon the idea of "success" and who defines what that word means. "The idea that others will be judging you to see how your students measure up on standardized tests causes many teachers to go against their better judgments of focusing on relationships and broader notions of learning to focusing on test preparation." (p. 43) Many of the theories mentioned, such as LatCrit Theory and Post-structuralism, question the relationship of knowledge and power and how it is used against marginalized and minority groups.

"The important point is that a sociopolitical perspective challenges whether the identities presented in the research project align with the ways in which educators and/or learners who are participants in the project would choose to describe themselves." (p. 45)

Gutiérrez also mentions the downsides to the sociopolitical turn and points out that it could be easy to begin to analyze and "fix" power and identity in a mathematics classroom, and lose track of the mathematics itself. "In the same way that I highlighted the importance of not focusing too strictly on mathematics so that social relations and advocacy disappear, we must also be cautious of not focusing on discourse to the point where mathematics disappears." (p. 56)

I agree with Gutiérrez when she points out that mathematics is in itself a subject that holds power over marginalized and minority groups, as the knowledge it imparted in educational institutions that not all members of society have access to. I also agree with her concern that often in the classroom it is easy to focus too much on the mathematics and neglect the relationships and culture held therein.

One of my thoughts when reading this article is that, looking back over the past few decades, education has had many “movements”, such as in the United States’ “No Child Left Behind” program, BC’s failed attempt at mathematics reform in the early 2000s, phonics vs. whole language learning, and others, that the sociopolitical perspective could be another. What I have observed in years past is that teachers and administrators will take on the new program or new ideology and focus all attention on it, leading to the neglect of other subjects or learning initiatives. The process reminds me of a see saw: we focus so hard on one topic that we ignore all others until there is a crisis, then we rush over to fix that, only to create yet another crisis in another area.

Do you think the sociopolitical perspective is a theory that you could easily integrate into your classroom and with your students? Or, would you want to? Do you find that you are frequently dealing with issues of power and identity in your classroom?

Using Two Languages

In the article, Using Two Languages When Learning Mathematics, Judit Moschkovich reviews research studies completed outside the mathematics field and applies them to mathematics, analyzing whether the information could be useful. 

Interestingly, Moschkovich writes that bilingual students are often slower than monolingual students with the retrieval time for arithmetic facts. However, the difference in calculation time was found to be minimal, between 0.2 and 0.5 seconds. The studies were also not taking in to account conceptual mathematics activities. "The results of these studies present a complex picture and appear in some instances to contradict each other." (Bialystok, 2001, p. 203)

The article goes on to also discuss "code switching", a term often disagreed upon, but generally referring to participants changing from one language to another during their part of a conversation. Sociolinguists have recorded that young bilingual children tend to speak in whatever language the conversation is begun in, whereas older students do not have this tendency as much. 

Overall, Moschkovich advises mathematics teachers to consider in their classrooms: what mathematical aspects are there? Is the work or problem conceptual or computational? What are the student's past experiences in each language? It is also important for teachers not to think of a student using code switching as being a deficit, as it is dependent upon context. It can be a helpful tool for the student to further explain their mathematical understanding. Bilingual mathematics learners have cognitive advantages, which could be further researched by focusing on how the bilingual learners communicate mathematically. 

Overall, I found this article to be mildly frustrating. It discussed many different studies and attempted to ground them by considering implications for a mathematics classroom, but most of the article called for more research. It outlined many areas where more research is necessary, so it did not come to many conclusions or make many findings. I did find it interesting that there is a difference, albeit small, between the computational abilities of bilinguals compared to monolinguals. It made me more curious as to why there is an impact in that area, as it is not often dependent upon language. I also wondered how I could use this knowledge in my teaching, as usually I do not speak the language that my students would be switching between. The article certainly made me reflect on my teaching and wonder how I could more carefully structure my lessons for all my learners, as I have many bilingual students. I certainly consider their competence with the English language when it comes to word problems, but I could do more when thinking about the required conceptual understanding of each unit as a whole. It did make me wonder: is this research relevant to teachers if the teachers cannot speak the second language of their students? 

Moschkovich writes, "mathematical discourse is more than vocabulary" (p. 138). Do you consider a student's bilingualism (or multilingualism) when choosing how to instruct conceptual understandings?  Would it matter whether the bilingual person was speaking a minority or majority language in their country of residence?