Final Thoughts

Normally I do not find text books over interesting; however, I have to say that reading Experiencing School Mathematics was not only interesting but engaging.   I thoroughly enjoyed the read and was disappointed to find out that Phoenix Park reverted back to traditional methods.

With such a great deal of research promoting open-ended project based instruction I think it is particularly sad that policy makers are not listening.  Just recently a new mathematics program was delivered to my province.  I do believe that this new curriculum is a step in the right direction, but it’s not where we need to be.  Students still feel disconnected with mathematics.  They have a difficult time understanding how it relates to them and how they can use it in their everyday lives.  If we are not teaching students how to connect what they are learning in the classroom to the real world, then what are we teaching them? 

When I was in high school I was very good at chemistry, it was one of my strongest subjects however, it did not interest me and I have not engaged in chemistry since.  If you were to ask me a question about chemistry now I would not be able to answer it, it would be alien to me.  I am not saying chemistry is not important, what I am trying to say is that mathematics is a valuable tool and we need to teach our students not just how to do it BUT how to USE it!! 

I believe this is the main message Boaler is portraying in her book and she reveals how through the use of project based instruction, students do get it.  They do know how to use mathematics in the world outside their classrooms!! 

I believe that this is the most valuable knowledge we can provide to our students.  

My thoughts on ability grouping…

I guess I am somewhat naive about ability grouping or tracking as some might call it.  I truly believed that ability grouping was a thing of the past, something my parents experienced in school, not me.  I was not taught in ability grouped classrooms, some students went out for remedial and that was the extent of it.  Nor, have I taught in ability grouped classrooms besides the streaming that occurs in the high school courses; general, academic or advanced streams.  I have heard of experiences however of some of my colleagues who were taught using ability grouping as young as elementary school.       

Boaler has certainly opened my eyes to this controversially subject in education.  Reading this week’s chapter allowed me to understand the negative experiences of students taught in ability grouped classrooms and researching this week for our weekly discussion and this blog has allowed me to realize just how popular ability grouping is. “The educational systems of most industrialized countries around the world use some form of achievement grouping (also known as tracking or streaming). [Furthermore, significantly] in many countries, achievement grouping has been the subject of heated political and scientific debate for many years now” (Trautwein et al., 2006, p. 788).

Advocates for ability grouping argue that placing students in sets allows teachers the ability to enhance the learning of higher achieving sets of students and lower achieving sets of students by meeting their needs within different learning environments.  Teachers teaching higher sets can spend more time on difficult learning activities and cover more material, while teachers teaching lower sets can spend more time on helping individual students meet the basic outcomes they need.  Boaler’s research flies in the face of these advocates in that it suggests that grouping by ability is doing more harm than good.  She is in line with many researchers who “have argued that placing an individual student in one of these clusters—commonly referred to as streaming, tracking, or ability grouping—will affect her or his achievement, future educational career, morale, and happiness" (p. 789).

Lucas’s (1999) work on tracking in the United States, “reported evidence suggesting that students in low-track classes received low-quality teaching in unsupportive learning climates and cited research indicating that placement in low tracks was associated with less favorable outcomes than placement in high tracks” (p. 789).  Boaler suggests something that is both in agreement with this and somewhat opposite to this.  She does demonstrate how placement in lower sets is detrimental to students’ success.  Once students are placed in lower sets their ability to achieve higher than the set they are in is almost non-existent.  She demonstrates the students’ feelings that they had “unfair restrictions [placed] on their potential mathematical achievement” (p. 164).  Also, along with Boaler’s conclusions there is an enormous amount of research that proves the negative effects on students’ motivation and self esteem when placed in lower sets.

I think it is important that Boaler demonstrates that students in the high-set were just as disadvantaged if not more than students in the low-set classes.  Through questionnaires given to students during her research at Amber Hill, who used the ability grouping approach, she illustrates Set 1 students at Amber Hill to be the most negative with regards to mathematics (Boaler, 2002, p. 161).  She asked students to respond to the question, whether they enjoyed mathematics lessons always, sometimes or never (p. 161). Students in Set 1 were the smallest proportion of students who responded that they always enjoyed mathematics, with not one single student saying they did (p. 161).  As well, they were of the greatest proportion of students who responded that they never enjoyed mathematics, with 27% stating this (p. 161).  Through the questionnaires and her observations Boaler summarizes that “the top two sets were made up of students who, at one time, were doing well in mathematics” and “despite this, the students liked mathematics less than other students and had less confidence in their own ability to do mathematics” (p. 161).   As she indicates, “for these students something had clearly gone wrong,” normally individuals who are good at math usually enjoy it (p. 161).  The students she interviewed and observed reiterated that they enjoyed grades 6 and 7 where they were taught and worked in mixed ability classes using an individualized approach (p. 161).  This is probably due to, as Boaler points out, that the top sets encompass an environment that has features of “rapidly paced lessons, competition among students, and pressure to succeed” (p. 159).  Students in the top sets at Amber Hill indicated that “the nature of their top set environment had diminished their understanding of mathematics” (p. 163). 

After reading chapter 10 and doing my own research, I agree with Boaler when she states that “placing students in academic groups often results in the fixing of their potential achievement” (p. 176).  Students in low sets feel like they are trapped and can never increase their achievement whereas many students in top sets break under the pressure.  In both cases, students love for learning and mathematical understanding suffer the consequences.

References:

Boaler, J.  (2002).  Experiencing School Mathematics.  New York: Routledge.

Trautwein, U., Ludtke, O., Marsh, H., Koller, O., Baumert, J.  (2006).  Tracking, Grading, and Student Motivation: Using Group Composition and Status to Predict Self-Concept and Interest in Ninth-Grade Mathematics.  Journal of Educational Psychology, 98 (4), 788-806.

Chapter 9

Myself and Shelly lead the discussion this week!

Different Forms of Knowledge & Different Mathematics Identities

Boaler ends chapter 8 with this statement: “the two approaches are not at opposite ends of a spectrum of mathematical effectiveness, but the differences between the approaches do serve to illuminate the potential of the different methods of teaching for the development of different forms of knowledge and the cultivation of different identities as learners and users of mathematics” (Boaler, 2002, p.136).  While Boaler uses this to end her discussion in chapter 8, I thought it would be a good place to begin mine. 

After sitting and pondering what I was going to write this week, I kept coming back to this quote.  I guess we could say that by the standards of the GCSE’s the two approaches are not at different ends of a spectrum with regards to mathematical effectiveness.  Both schools had relatively similar exam marks.  However, we must keep in mind in our society, mathematical effectiveness is measured in terms of exam marks and therefore the different approaches are comparable in that regard.

However, I think Boaler sheds important light on the fact that the different approaches to instruction supports the development of different forms of knowledge.  Amber Hill students were taught in a traditional manner and their knowledge was strictly procedural.  Students had “learned the teachers’ methods and rules without really understanding them” (p.122).  Whereas, the students at Phoenix Park who had been taught in a reform setting had developed knowledge that was internal and all encompassing, “students learned to adapt and change methods to fit the demands of different situations” (p.125).  Students at Phoenix Park had “developed a usable form of mathematics in response to their project work,” unlike Amber Hill students who seemed lost and disconnected from real world mathematics.  Students at Amber Hill “reported that school mathematics was unrelated to the mathematics problem they encountered in their jobs and lives, and when they want to use methods many of them reported they were unable to apply them” (p.123).  If there were no cues for students to follow they were incapable of continuing with the problem or they could not interpret the demands of the question correctly.  Boaler refers to the Cognition and Technology Group at Vanderbilt study as “they report that problem-oriented approaches to learning help students view mathematical concepts as useful tools that they can use in different situations.  More traditional approaches to learning cause students to view concepts “as difficult ends to be tolerated rather than as exciting inventions (tools) that allow a variety of problems to be solved” (p.123).  From this one must question what type of knowledge is most valuable?  Students with procedural knowledge but cannot apply what they know within their real lives? Or, students who may not know every procedure but have inherent mathematical reasoning available to them to navigate their way through a problem? 

Both sets of students cultivated different identities as learners and users of mathematics because of the approaches to instruction that was integrated within each school.  The way each set of students “connected and interacted with mathematics and formed mathematical relations was different because of the practices in which they engaged in school and the effect of those practices on the mathematical identities students developed” (p.135).  Students at Amber Hill “did not regard themselves as mathematical problem solvers.  [Meaning] that when they met new situations, they abandoned the mathematics they had learned in school and were forced to rely on their own invented methods” (p.130).  Students had a general opinion that mathematics was fixed and was only relevant in their mathematics classroom.  These students were set in the novice stage of learning, given little opportunity to gain an understanding of the mathematics they were learning.  This is different from students at Phoenix Park who “had engaged as mathematical problem solvers, and they were willing to consider any mathematical situation and think about what was involved” (p.134).  Students had “developed more effective forms of knowledge and more productive identities as mathematical learners” (p.134).  Students saw no boundaries with regards to mathematics, giving them more access to their school-learned mathematics in their real lives.  The students in our classrooms will be leading us into next generation; do we want students taught in traditional settings who are “received knowers-expecting to accept knowledge from authoritative sources without contributing to knowledge or making connections across areas”? (p.122).  Or do we want students taught in progressive classrooms who are empowered by mathematics, who identify themselves as mathematical problems solvers, and who have “developed a predisposition to think about and use mathematics in new and different situations”? (p.126).

Just some thoughts!

References:

Boaler, J.  (2002).  Experiencing School Mathematics.  New York: Routledge.

"Dan Meyer: Math Class Needs a Makeover" & Chapter 7

This week while I was researching traditional and reform classrooms and real life application to do my blog post concerning the issues in chapter 7, I came across a video that I think has an imperative message to educators today.  So instead of writing about chapter 7 explicitly, some of the topics I was going to discuss are addressed in the video, so I am going to discuss the video instead, with hopes that everyone will view it. 

On the TED website I viewed the video, “Dan Meyer: Math Class Needs a Makeover.”  In the video Dan Meyer discusses classrooms in the United States today, and I could not help but draw the resemblance to that of Amber Hill’s classrooms, which Boaler describes in her book.  He outlines examples of how the current curriculum, text books and societal values are diminishing any chances of students being able to become patient problem solvers and efficient in mathematical reasoning.  He then goes further by describing a solution to this.  The instruction methods he illustrates are similar to those Boaler describes in her portrayal of Phoenix Park’s classrooms.  Meyer gives an even clearer picture of how we need to challenge students with real life applications and use the technology we have at our finger tips as aids to produce a generation of real mathematical problem solvers.

I think the video is exceptionally interesting and takes Boaler’s description of Phoenix Park’s approach into the 21^st Century.  For myself, it gave me a clearer picture of how the teachers at Phoenix Park developed and approached problems they introduced to students, and how they allowed student discussion to generate the right questions and lead to solutions.  He ends the video with statement:

Math makes sense of the world.  Math is the vocabulary for your own intuition.  So I really encourage you, whatever your stake is in education...insist on better math curriculum.  We need more patient problem solvers" (Meyer, 2010).

I hope everyone takes a moment to view the video, I think it is an excellent for professional development piece!

Video: Dan Meyer: Math Class Needs a Makeover  

As well, I have attached the link for his blog below.  On it there is a wealth of resources for teachers, hope you get a chance to have a look.

Dan Meyer Blog


References:

Boaler, J.  (2002).  Experiencing School Mathematics.  New York: Routledge.

Meyer, D. (2010).  "Dan Meyer: Math Class Needs a Makeover."  Retrieved from http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html on November 5, 2011.

Situated Learning

In chapter 6, Boaler delivers an account of the variety of assessments she investigates to observe “whether differences existed in the extent, nature or form of students’ understanding” (Boaler, 2002, p.84).  Within these she develops two applied assessments which further her aim to investigate the notion of situated learning.  “Lave and Wenger (1991) argue that learning should not be viewed as simply the transmission of abstract and decontextualised knowledge from one individual to another, but a social process whereby knowledge is co-constructed; they suggest that such learning is situated in a specific context and embedded within a particular social and physical environment (p.40)” (Wikipedia, 2011).  At Amber Hill students have very little exposure to situated learning, unlike Phoenix Park students who are completely immersed in it within their classrooms. 

Through the applied assessments Boaler makes a number of interesting observations about the students from both Amber Hill and Phoenix Park.  In order to test students performance, at two different points in time Boaler provides two activities: the architectural activity and the planning a flat activity.  As students in both schools completed these activities some significance differences regarding students’ mathematical understanding arise. 

Before beginning both activities students had to complete related questions concerning the concepts that would be needed to solve the activity.  In both cases, Amber Hill students scored higher.  However, while completing the activities Phoenix Park significantly outperformed Amber Hill students.  During the first activity, although students at Amber Hill were taking from the top set (meaning they were the strongest academically in that cohort), “students related to an inability to decide what to do when they were not given explicit instructions” (p.88).  Even though students had learned the appropriate knowledge to solve the problem, when they had to make decisions as to what methods to chose to solve the problem they became lost (Boaler, 2002, p.88). 

The second activity provided another insight into Amber Hill’s lack of mathematical understanding.  While completing the producing a flat activity Amber Hill students showed difficulty in interpreting the situation and the goals and demands required of them.  Students did what they thought was required of them, “ignoring the situation or context in which they were placed” (p.92).  Students had the appropriate knowledge to solve the problems but seemed unable to interpret what they had to do; having “difficulty making use of the mathematics they learned in an applied situation” (p.93).    

Finally, during the planning a flat activity students at Phoenix Park illustrated a significant degree of creativity when planning their flats.  Their designs included unusual rooms and “were ingenious, entailing a creative use of space with interlocking rooms that saved on redundant hall or corridor space.  In effect, the students often gave themselves a more demanding cognitive task, but managed to attend to the given rules and constraints of size and scale to produce impressive designs” (p.91).  Students from Amber Hill showed much less creativity with designs that were “inaccurate, sketchy, and basic”, even though students seemed to really enjoy the activity (p.91).

I think Amber Hill is an example of a school where “when learning is removed from its context, the value of the knowledge and the relevance of that knowledge to the learner become depreciated” (Duffy & Cunningham, 1996).  As Boaler points out students from Amber Hill had difficulty making use of their mathematics, not “due to a lack of mathematics knowledge, but the ways in which students interpreted the demands of the activity” (p.93).  Learning is a social event, and in a classroom where students are expected only to regurgitate information instead of think for themselves it cannot be expected that quality learning will ensue.  Even though students show much enthusiasm and motivation for the activity, they had not developed the skills to be successful.

Students at Phoenix Park showed much success with both applied assessments.  The instruction they received throughout their mathematics education made them bettered prepared to use their mathematics skills, understanding and interpreting the problems they faced.  

References:

Boaler, J.  (2002).  Experiencing School Mathematics.  New York: Routledge.

Duffy, T. & Cunningham, D. (1996). Constructivism: Implications for the Design and Delivery of Instruction. In D. Jonassen (Ed.), Handbook of research on educational communications and technology. New York: Simon & Schuster.

Lave, J. & Wenger, E.  (1991).  Situated Learning.  Legitimate peripheral participation, Cambridge: University of Cambridge Press.

Wikipedia.  (2011).  Situated Learning.  Retrieved from http://en.wikipedia.org/wiki/Situated_learning on November 5, 2011.    

The uninterested student and the reform classroom

After reading chapter 5, I have to question would the students who seemed to be off task be uninterested in any classroom, not just the open-end mathematics classroom at Phoenix Park? 

The following is an excerpt from the text on page 75:

JB:          Why do you mess about in maths more that other subjects?

S:            Because half the time if I ask for help I don’t get it, or I don’t get it until 20 minutes after I’ve asked. (Shaun, PP, Year 10, RT)

In a class of 30 students I will willingly admit I have a hard time getting to every student.  With 30 plus students in a classroom, one on one time is limited.  Some students do have to wait 20 minutes.  This is a sad reality to our current system with no cap sizes on senior high classes.  Like some of my classmates have noted in their discussion postings, they gravitate to the students who they identify need their help but won’t ask, I do this as well.  I spend a lot of time with students who if I didn’t ask would never ask me for help, they would just sit there.  In doing this I sometimes fail the student who excels in the subject because they are independent enough to engage in their work themselves.  As a result, I am failing my academically proficient students because I am not allowing them to achieve their highest mathematical levels.  I do provide my students with enrichment activities as much as possible, however, I really feel this is not enough. 

Finding a balance is extremely difficult with the number of students in our classrooms and time restraints, and it changes every year with new students and new atmospheres.  There seems to be no perfect answer.  If someone has an answer please let me in on the secret…