Final Reflection

Well, we’ve come to the end of the term and we’ve completed the reading of Boaler’s study of Phoenix Park and Amber Hill. Boaler  hopes that her research “has furthered understandings of the relationship between different classroom interactions and the understandings, beliefs, and dispositions students develop” (p. 182). She states that she is not implying that Phoenix Park represented and ‘ideal learning environment’ but if the lessons were improved it would not be moving towards the Amber Hill model.

Phoenix Park teachers had high expectations for all students, allow students to think for themselves, to interpret mathematical situations, choose methods, and solve problems. They were able to use the math they learned outside of the classroom and the activities they did inside the classroom were more meaningful to real life. The activities were not just procedures and rules to learn and follow.

In beginning this course and the reading of the text, I had an understanding that activities in the classroom needed to be meaningful and related to real life. As I read through the text and other articles throughout the term this understanding has been strengthened.

Eisner, as cited by Flinders &Thornton (2009) points out that the activities that youngsters take part in within the classroom promotes the way they think, and if they have no reason to raise questions, the processes that help them learn how to discover intellectual problems are not developed. Students need to make connections between what they study in class to out of class. According to Eisner, there has to be a ‘transfer of learning’. Students need to be able to apply what they have learned and engage in the kind of learning they will need in order to deal with situations outside of the classroom. “There is a difference between what a student can do and what a student will do” (Eisner, p. 331).

Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to

      teaching and their impact on student learning(Rev. and expanded ed.). Mahwah,   

      NJ: Lawrence Erlbaum Associates, Inc.

Flinders, D. J., & Thornton, S. J. (2009). The Curriculum Studies Reader. 3^rd edition. New York, New York:

      Routledge.

Strategy to use in Mathematics Classroom

In researching for my Inquiry Project I came across a teaching strategy that I believe is worth sharing. In the article, Strategies For Teaching In Heterogeneous Environments While Building A Classroom Community by Hayley Lyn David & Robert M. Capraro () they discuss a strategy developed by Zemira R. Mevarech & Bracha Kramarski called IMPROVE. This strategy involves the following steps:

1)      Introducing new concepts: teacher introduces concept to the whole class.

2)      Metacognitive questioning: students are placed in small (2-4 students)heterogeneous groups to engage in metacognitive questioning (they take turns asking 3 kinds of questions: i)comprehension(articulate main ideas in problem, classify problem in appropriate category, and elaborate the new concepts; ii) strategic (questions that refer to appropriate strategies for solving the problem  and, iii) connection (questions that often refer to similarities and differences between the posed problem and other problems previously solved.

3)      Practicing: students cooperatively work to develop and agree on a working solution.

4)      Reviewing and reducing difficulties: through questioning the students review and reduce difficulties; students worked from different perspectives and checked their work to find the simplest solution.

5)      Obtaining mastery: as students practice cooperative learning through metacognitive questioning, each student mastered each concept at various rates.

6)      Verification: along with teachers’ observations and informal assessment, the students are given a formative test to verify mastery.

7)      Enrichment: students who master the concepts move on to enrichment tasks.

David, H. L., & Capraro, R. M. (2001) Strategies for teaching in heterogeneous environments

     while building a classroom community. Education, 26, (1), 80-86.

Girls/Boys/Learning Styles/Gender Equity.........

In chapter nine Boaler discusses Boys, Girls, and Learning Styles.  Boaler states that the ‘greatest disadvantages were experienced by the girls mainly because of their preferred learning styles and ways of working” (p.137). She contends that girls from the highest sets at Amber Hill underachieved because of the learning approach the school took. She stated that these girls “seemed to value aspects of mathematics teaching and learning that were not present in their school’s approach” (p. 149). These girls at Amber Hill wanted to ‘understand their mathematics’ (p. 153). Boaler stipulates that mathematics is being widely taught in a way that is not equally accessible to boys and girls, and this appears to relate to the preferences of pedagogy (p. 152).

According to Anthea Lipsett a writer of education issues, formerly for the Education Guardian, “Boys are not innately better at maths than girls, and any differences in test scores is due to nurture rather than nature”, according to researchers (Lipsett, 2008). She quotes research by Prof Paola Sapienza of Northwestern University in the US as saying “The so-called gender gap in math skills seems to be at least partially correlated to environmental factors, the gap doesn’t exist in countries where men and women have access to similar resources and opportunities.” According to data analysed by researchers in 40 countries, boys did tend to outperform girls in math, but in more ‘gender equal societies’ such as Iceland, Sweden and Norway, girls scored as well as boys or better. The research found a striking gender gap in reading skills. In every country girls performed better than boys in reading but in countries that treat both sexes equally, girls do even better.

Paul Blundin (2009) states that just about every scientific instrument in the world have come to a fairly inescapable conclusion that boys and girls learn differently because their brains develop differently. His article says that boys and girls use different learning intelligences to gather information but although it may take effort and stimulation for a brain that has a strong spatial bias to develop its more logical-mathematical abilities, but it can be done. Blundin claims that although boys generally are more capable in the logical-mathematical category this is changing and girls have been gaining ground in this area. He attributes this to how society is desiring to encourage girls in math. Blundin writes, “Cooperative learning experiences that involve more active tasks than just writing, show girls attending to the task more readily and socializing more productively. Boys want to get to the project and get moving and doing”. This comment compares to Boaler’s findings that Phoenix Park girls performed better than Amber Hill girls because at Phoenix Park the girls were free to develop their own styles of working, encouraged to think for themselves, discuss ideas with each other , and work at their own pace. (p. 148)

Blundin ends his article with “Those dedicated to teaching… must see every child as a potentially multi-intelligent child and provide as much stimulation in all the areas as possible.”

I don’t know if I grew up in a home, community, and school who treated boys and girls equally when it came to learning mathematics or not but I don’t remember ever thinking that I couldn’t or shouldn’t be better at mathematics than boys. 

http:www.guardian.co.uk/education/2008/may/30/schools.uk1

http://old.eduguide.org/Parents-Library/Learning-Intelligences-Gender-Behavior-150.aspx

Mathematical Identities

Reading through the text book up to and including chapter 8, I became intrigued by ‘mathematical identities’. It’s interesting how the attitudes and beliefs of the students at both Amber Hill and Phoenix Park influenced how they were able to use math in different situations. As I dug into the topic a little more, I came across a discussion article by Grootenboer and Zevenbergen (2008).

 Grootenboer and Zevenbergen state that it is important to have this discussion about mathematical identities because there is a developing problem with students not being engaged and not participating in mathematics and there is a concern about mathematical achievement as well as poor attitudes toward mathematics (p. 244). In this article the authors discuss how students learn mathematics and continue to develop a mathematical identity through a classroom context. They discuss the three main aspects of the classroom as being the student, the discipline of mathematics, and the classroom community.

The student brings to the classroom community an already developing mathematical identity that has been formed by engaging with family, peers, etc. This identity has already been influenced by ‘their previous experiences of mathematics education’ and these experiences will affect their future learning in mathematics (p. 244). The teacher is a significant feature of the classroom community (p. 245)  but the community also includes the other students and the physical environment.

The authors stipulate that ‘if the goal of mathematics education is to develop a strong mathematical identity, then the critical focus is the relationship between the student and the discipline of mathematics’ (p. 245) and the ‘facilitating context for the development of this relationship is the classroom community, and specifically the teacher’ (p. 245).

Grootenboer and Zevenbergen  discuss that the classroom community is only temporal , and the only thing that will last is the mathematical identity which is the connection between the student identity and mathematics.  The teacher is the bridge between the student and mathematics. The teacher must be knowledgeable in the area of mathematics and must have a well-developed mathematical identity himself/herself. The teacher should have ‘a positive attitude towards the subject, a sense of joy and satisfaction in undertaking mathematical practices’ (p.246). The teacher must also have a relationship with the students because it is the teacher’s role to facilitate the development of students’ mathematical identity thus bridging student and subject (p.246).

From reading this article and the text chapters in Boaler’s work (2002) I have a better understanding of how students develop their attitudes and disposition towards mathematics and how I (as the teacher) have a leading role to play in helping to develop my students’ mathematical identities. It is very important that I seek professional development in mathematics to stay on top of new learning trends and teaching ideas, to give my students a positive outlook on mathematics, and most important develop a relationship with my students as we learn and enjoy math together. I only have them for one year and then they take their mathematical identity with them to another community classroom where they will continue to develop their identity further.

Boaler, J. (2002). Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and their Impact on Student Learning. Lawrence Eribaum Associates: Mahwah, New Jersey.

Grootenboer, P. J., & Zevenbergen, R. (2008). Identity as a lens to understand learning mathematics:

Developing a model. In M. Goos, R. Brown, & K. Makar (Eds.), Navigating currents and charting

directions (Proceedings of the 31st annual conference of the Mathematics Education Research Group of

Australasia, Brisbane, Vol. 1, pp. 243-250). Brisbane: MERGA

http://www.merga.net.au/documents/RP262008.pdf

Janna and I are responsible to lead discussion for the week of Nov. 9th  to 15th.

Jean Lave and situated learning

In chapter 6 Boaler states that one of her aims in this study was to investigate ‘situated learning’ and she cites Lave in her research throughout chapters 6 and 7. Jean Lave is a social anthropologist with a strong interest in social theory. She teaches geography and education at the University of California, Berkeley. A lot of Lave’s work in situated cognition and communities of practice has been done with Etienne Wenger. Lave received her Ph. D. in Social Anthropology from Harvard University in 1968.

In 1991 Lave worked with Wenger to propose the situated learning model of learning. This theory suggests that learning involves social interaction, and that knowledge should be presented in an authentic context. In one of Lave’s projects she compared the way ‘just plain folk’ learned to the way students learned in the classroom and she found that apprentices experienced great learning success through authentic activity without actually being taught to. This is similar to what the teachers at Phoenix Park School were trying to do. They gave students projects that were open ended and similar to real life and allowed the students to work on these projects without actually teaching them the skills and concepts.

When designing learning experiences from the situated learning perspective, one believes that knowledge is acquired through situations and this knowledge is transferred only to similar situations. That the knowledge gained out of context is more difficult to generalize to an unfamiliar situation. Students at Amber Hill seemed to have run into this difficulty. When they were asked to answer a question or solve a problem that was unfamiliar to them, they had difficulty applying the math they had learned in isolation.

Looking at learning from a situative perspective means looking at the group of learners as more important than individual ideas. The individuals in a group consider, question, and add to each other’s thinking so that important mathematical ideas and connections can be produced as a group (Brodie, 2005). According to Brodie (2005) situative perspectives argue that what a learner says and does in the classroom will make sense from the learner’s perspective of knowing and being, from the learner’s identity in relation to mathematics and to the learner’s past experiences of learning math, both in and out of school. She states that if learners have a particular expectation of ways of working in math classrooms and of what counts as appropriate contribution in the classroom, that they will continue with this outside of the classroom.

Situated learning theory seems to apply more to Phoenix Park School where the students are given freedom to work with others at their own pace, using their own constructed ideas and skills to solve relevant, and authentic problems. When they learn this way they are more able to transfer this learning to other situations. Amber Hill School on the other hand seems to be following a more traditional learning method where students are given closed end questions to work on individually using learned formulas and procedures which they find difficult to transfer outside of classroom use.

“Situative perspectives argue that a focus on conceptual structures is not sufficient to account for learning. Rather, interaction with others and resources are both the process and the product of learning and so learning cannot be analysed without analysing interactional systems.” (Brodie, 2005)

References

Boaler, J.(2002). Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and their Impact on Student Learning. Lawrence Erlbaum Associates:Mahwah, New Jersey.

Brodie, K. (2005). Using cognitive and situative perspectives to understand teacher interactions with learner errors. Retrieved from the internet October, 2011.

http://www.lifecircles-inc.com/Learningtheories/constructivism/Lave.html

http://www.ischool.berkeley.edu/people/faculty/jeanlave

http://elpea.tripod.com/jlavebio01.html

GAMES

Games – Just a fun fill in OR an effective instruction and/or assessment activity?

Just last week I had a substitute teach my class for a day while I was away.  The sub worked on equivalent decimals and renaming decimals as fractions. When I returned the next day I realized that after this day my students would have a long weekend and because I wasn’t in class yesterday I didn’t know how well they had grasped the decimal concept. I decided to have them play decimal snap so that I could circulate and get an idea of their understanding of what an equivalent decimal was. Each pair of students was given a stack of cards which had either a decimal number written in the tenths, hundredths or thousandths or a picture of a grid divided into tenths , hundredths or thousandths with part of the grid shaded. The only instructions I gave the class was to share the cards equally with your partner, each person turn over a card and if you get a pair of equivalent decimals say ‘snap’.

Circulating the class it was so much easier to get a grasp on who understood the concept than it would have been if I had given them written text book questions and had taken these in to correct one by one. An added bonus was that the students loved math class and had a ball reviewing and learning a math concept.

A couple of things I noticed that I had not planned to assess or ‘look for’ happened that I found interesting.

1)      A student of ‘lower ability’ became bored with the game and decided not to play after a few minutes. She spent the rest of the class just ‘wandering’ while others played. This was similar to the students at Phoenix Park who were off task.  I wonder, would I allow this to happen in all my classes if this student decided not to participate? I wonder how my administrator would react to this?  I don’t think this would fly as a continual practice in elementary especially and not likely in high school at my school.

2)      I was pleased with the number of different ways my students chose to play the game. As I circulated the 12 groups I noticed that about half of the groups had devised another method of play. Some had combined rules from other games they had played such as ‘go fish’ and others had decided to only deal out half the cards and keep a center base pile. In the end it didn’t matter how the played the game, only that they were reviewing and learning about equivalent decimals while having fun.

Boaler, J.  (2002). Experiencing school mathematics:  Revised and expanded edition.  New York:  Lawrence Erlbaum Associates Inc.