Gender issues in Mathematics

Gender in Mathematics presentation

Algebraic Thinking in the Early Years

In Carolyn Kieran’s reaction article on theories of algebraic thinking, she discusses a model with 3 components:

1. the generational activities of algebra-rules governing numerical relationships, variables and unknowns, the equal sign and the equation solution.
2. The transformational (rule-based) activities-collecting like terms, factoring, expanding, substituting, etc.
3. Global, meta-level, mathematical activities-problem solving, modeling, change, relationships

In the examination of different approaches on preparing students for algebra, Kieran goes on to note there is no international consensus on what is algebraic thinking and how best to teach it. She mentions the Chinese curriculum, where its goal is to understand quantitative relationships as well as the Singapore primary mathematics program which focuses on problem solving and relationships. The Korean curriculum focuses on relationships and structure as well as problem solving. The U.S. seems to stand a bit apart, instead focusing on number patterns and generalizations.

Kieran does point out the commonalities between the programs in the development of algebraic thinking: relationships, generalization, justification, problem solving, modeling and structure.

I found Kieran’s article to be insightful and agreed with many points. I found it interesting that many of the curricula in Asia were similar, in that they focus on problem solving and relationships. I have certainly noticed a difference between the mathematical abilities of students who have used the textbook I’m most familiar with, Singapore Math, and students who have come from schools using different texts, such as Math Makes Sense, without such a strong focus on operations and understanding.

I also didn’t consciously realize that there could be students operating with an arithmetic frame of mind and those with an algebraic way of thinking in my classroom. After Kieran’s description of the difference, I feel I could confidently identify most of my students as one or the other.

In several sections of the article, Kieran mentions the importance of the equal sign and that students understand its meaning. It reminded me of an activity another teacher once did with a group of elementary teachers: 11 + 4 = ____ + 3 = _____ + 7 = _____. Most teachers said the answer is 25!

Kieran mention 5 items that should be focused on to succeed in algebra: relations, operations and their inverses, representing and solving problems, numbers and letters together and the meaning of the equal sign.” In your experience, do you think students have a good understanding of these concepts? Or do you find you need to review/re-teach some of them?

Embodiment in Mathematics

In Embodiment in Mathematics Teaching and Learning: Evidence From Learners’ and Teachers’ Gestures, Martha W. Alibali and Mitchell J. Nathan present research that shows how the gestures teachers and learners use show mathematical knowledge. Specifically, they discuss 3 types of gestures, which occur in instruction and explanations:

1. Pointing gestures, which link speech to the physical environment.
2. Representational gestures, which include a person, teacher or learner, using body movements to communicate a specific meaning.
3. Metaphoric gestures, which are visual representations of abstract ideas.

The researchers mention that pointing gestures were by far the most commonly seen types in elementary classroom, likely because of the prevalence of manipulative in those classrooms. All of the types of gestures were used by speakers when thinking and explaining mathematical ideas were thought to be intentionally produced to facilitate communication.

Alibali and Nathan conclude “embodied knowledge is an integral component of mathematical thinking and learning. Gestures thus provide a unique and informative source of evidence regarding the nature of mathematical thinking” (p. 274) They believe that gestures should be implicitly taught in teacher education programs, as they are as important as managing and assessing a classroom.

I had several stops in this article. One of the parts of their research that I wondered about was how the data was collected. Were they recording classes? They must have been, but there is not much discussion on this point. I was also wondering how many teachers and learners they studied and at what ages. They do discuss how elementary teachers use pointing gestures more frequently than the other two types, but what is the difference? Is it significant? I would assume so, as it is mentioned as an important fact.

I found it interesting that elementary teachers used pointing the most often. Certainly I can appreciate that it makes sense that they wouldn’t use the metaphoric gestures as frequently, as they are connected to abstract ideas and mathematics-based abstraction is only introduced at the end of elementary. But I was a little surprised that elementary teachers didn’t use representational movements more. I think of these movements as a way to connect the concept to the students. When you get students moving and doing, I have found they are much more likely to learn and remember the concept in the following year.

As the article mentions, it provides compelling proof that gestures are very important to cognition in the mathematics classroom. It made me wonder whether most teachers are aware both of the importance of gesturing, but also if they are doing the gesturing consciously or unconsciously. Are learners gesturing only if they have had a teacher who consciously uses gestures in the classroom?

Do you see gestures as being an integral part of your teaching? Upon reflection, do you think you use one type of gesture more frequently than others? Or are you not aware of the gestures you use?

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?