Looking to the Future is the summary of Boaler's research on the two schools, Amber Hill and Phoenix Park, where she related the differences in teaching and learning found there. Boaler's book makes a very strong case for the need for reform in mathematics instruction away from the traditional form of instruction (like that received at Amber Hill) toward an inquiry-based, problem-solving approach ( like that at Phoenix Park). Boaler states that remembering facts or procedures for a test does not signify mathematical understanding and we have, over the course of this book, seen that this is true. .
When the students of Amber Hill were asked (in the GCSE's) to apply mathematical knowledge to new situations they were unable to do so. This was largely due to the traditional form of instruction they had received, the cue-based learning they had experienced and a dependence upon the teachers providing additional structure whenever frustration was met in class. We, have as a group of teachers in our class, commented many times on the 'wonderful' experience of realizing students cannot recall concepts you worked on even a short time ago. Teachers have always asked "What's wrong with them[students], we just covered this?". Well, now we know that what's wrong with them is they didn't achieve true understanding. They could memorize steps or repeat examples of something the teacher had just modeled, but as far as really seeing what the mathematics was in a given topic, that was something not achieved by many.
Boaler makes a strong case for the "inherent complexity of the learning process" and for the idea that we cannot see assessment as an indicator only of whether or not "a student has more or less knowledge". Assessment is so much more than that and as a teacher who was already making changes to her assessment style I have been profoundly affected by what I have learned from Boaler's work. I was indeed making changes to assessment but I didn't truly understand the need for it, I didn't have the belief that it would make a difference. Now I know that assessment must be focused on finding out about the different forms of knowledge a student has, the depth of mathematical understanding and reasoning a student develops, the connections they make to the big ideas of mathematics and most importantly assessing how these understanding, reasoning or connections were made.
This chapter made some very interesting conclusions and I was most struck by the following ( this is by no means a total list of the ideas contained within the pages of this chapter):
-Pedagogies matter. Simple statement, profound idea. If nothing else this fact has been hammered home to me during this class, my teaching has been affected for the good by this.
-"if we want student to consider mathematical situations and flexibly make use of mathematics knowledge in the real world or in examinations of higher mathematics, we need to engage students in similar practices in the classroom". As we discussed, I know I have made changes to my teaching style but I don't think I have made the situated learning in my classroom relate to the real world in any worthwhile way. Something to work on!
-'spoon feeding' students is not helpful. This is certainly something we've discussed at length and I am as guilty of it as anyone of doing this. I am more aware now and make a conscious effort not to enforce learned helplessness by becoming a crutch for students. I am more willing to let them 'struggle' and try to find the answer their own way.
-The need for professional development is paramount. I did choose this as my topic of inquiry and I've learned a lot from that research and this book. When Boaler states on p. 182 that " if similar energies [she refers to the lengths( and money)anti-reform lobbiests go through] were spent helping teachers become better prepared then it seems likely that the nations children would be considerable better served". Yes, this is exactly my point, I can't see reform measures in mathematics taking hold in a big way without significantly more support for teachers ( time, money, learning situations).
This study provided many critical issues for mathematics instruction for us to delve into. It was an interesting book and an enjoyable course. I was rather sad to read in the end that political/parental pressure has scuttled the innovative work that was being done at Phoenix Park. I don't believe that throwing money at the problem is going to help improve understanding. Standardized tests scores going up ( which has not been the case) do not mean the children in Britain ( or anywhere) have a better grasp of mathematics. The advantages of the approach taken by Phoenix Park have been substantiated by Boaler's longitudinal study, too bad it seems to be so easily ignored. Nevermind, it has changed at least 8 people we know and that's a start! Thanks to everyone for such an enjoyable semester, good luck with your future studies, thanks to Mary for her help and guidance and Merry Christmas to all!
Monday, December 7, 2009
Saturday, December 5, 2009
Wednesday, December 2, 2009
Ability Grouping, Equity and Survival of the Quickest
The use of ability grouping is certainly a contentious issue in education. Should students be taught in groups where they are with students of similar ability( homogeneous groupings), being taught at a rate that is designed for their needs. Or, should students be taught in mixed-ability (heterogeneous groupings) where some will find the pace and material challenging but will allow them more opportunities for post-secondary education? In Newfoundland and Labrador all students are taught in heterogeneous groups until they reach junior high, where, for some subjects such as mathematics, they are tracked into more homogeneous groupings based on ability. This topic has come up several times in our discussions as a group, as an elementary teacher it never occurred to me to question what 'tracking' really means to a child's future. Why would education systems track students according to ability? What exactly is at stake when we use ability sets to teach students, how are their needs best met, how can they be motivated to learn, what are the implications for their future, what is the overall effect of being tracked? The importance of the answers to these questions cannot be overstated, Boaler's findings in this chapter go a long way to answering these questions for me.
One of the main issues involved is what are the advantages/disadvantages of being placed in 'like' ability groups for mathematics or in other words, why do we use this type of grouping at all? According to Boaler the main reason is "so that they [the teachers] can reduce the spread of ability within the class, enabling them to teach mathematical methods an procedures to the group as a unit. They "cannot see how "bright" students may may be stretched or "weak" students supported" (p. 155) when a mixed-ability grouping system is used. Tracking students by ability allows the teacher to give the form and speed of instruction that is most useful for those students appears to be the basis of the argument for this type of education system. Students expressed two main complaints or concerns with this type of instruction they received, one was that 'fixed pace diminished understanding' and the other was "the pressure they felt was created by the existence and form of their setted environments". The need to work at a paced fixed by the teacher was incredibly frustrating for students, particularly in the higher sets where the pace was the fastest, " I don't mind maths but when he goes ahead and you're left behind, that's when I start dreading going to maths lessons (Helen, AH, Year 10, Set 1)". When in homogeneous groupings students also felt they were constantly being compared to and judged by peers. While the basis of using homogeneous groupings is that it would motivate students and raise achievement this was found by Boaler to be true for a very small number of students.
The use of homogeneous grouping has some inherent components which are detrimental to the learning of students in the lower sets. What is at stake when we track students according to ability is perhaps the most disturbing finding of all in this chapter. Boaler refers to this as " Restricted Opportunities" students were angered by 'what they felt to be unfair restrictions on their potential mathematical achievement". Students were if anything dis-motivated to learn by tracking restrictions "they could not see the point in working in mathematics for the grades available to them" (p. 166). Students often felt that the set they were put into was not fair that the decision was based on items other than ability such as personality clashes between them and teacher or on their behaviour. Boaler uses information from Tomlinson (1987) which provided evidence that student behaviour does in fact play a role in the decision about which set they were assigned to. To me,subjectivity of the teachers feelings towards a student should not be allowed to play a part in the decision behind where that student ends up! In her summary Boaler states that 'setting' did not affect all students the same, some were indeed given an advanatage but others were negatively impacted. The consequences of being setted, especially those in lower sets was important to students and was discussed in greater detail later in the chapter.
When Boaler looked at the mixed -ability groupings at Phoenix Park and compared them to Amber Hill she found that AH students had something to react against that the PP students did not because grouping was not an issue at the latter school. The teachers at PP chose activities that gave access to students "at multiple levels" of ability, this is of course related to the newest train driving education in Newfoundland and Labrador ( and elsewhere) otherwise known as Differentiated Instruction(DI). I have to admit I don't have much experience with DI , I attended two workshops on this topic both of which were way above my level of ability as they were designed for teachers who had been working with this system and had experience using it components. These experiences left me feeling frustrated and overwhelmed and so I just used what I could glean from them in my own fashion. After reading this chapter, I am once again reminded how useful DI can be and will probably delve into a little more being careful to begin at a place where my needs are met. Obviously students at AH who, in a set , are all being taught the same material at the same pace are not having their needs met. When Boaler compared the achievement levels between AH and PP students her research found that mixed ability grouping increased achievement opportunities for all students, particularly girls and low socioeconomic status students, which lends itself to the issue of equity in mathematics.
I was profoundly disturbed by Boalers revelation that "students are rarely told the implications of the group withing which they are placed." This is shocking to me, if the knowledge of the group they are placed in must be kept secret because it is detrimental to motivation why wasn't that a flag that perhaps a closer look at this system was in order. The short term and long term effects of tracking are enormous. In the short term students go through school being compared or comparing themselves to others in an unfavorable light. Students who work at a lower level and don't know it and who are achieving well are shocked ( and understandably so) that they haven't achieved what they thought. In the long term, where you are placed at 13 determines for many the outcome of their professional or adult lives! Imagine thinking you were doing really well in math only to find out that you weren't really learning what was needed to give you the greatest options for post-secondary education, yikes! I can't believe this has been allowed. This was yet another eye-opening chapter and one I am glad to have met.
One of the main issues involved is what are the advantages/disadvantages of being placed in 'like' ability groups for mathematics or in other words, why do we use this type of grouping at all? According to Boaler the main reason is "so that they [the teachers] can reduce the spread of ability within the class, enabling them to teach mathematical methods an procedures to the group as a unit. They "cannot see how "bright" students may may be stretched or "weak" students supported" (p. 155) when a mixed-ability grouping system is used. Tracking students by ability allows the teacher to give the form and speed of instruction that is most useful for those students appears to be the basis of the argument for this type of education system. Students expressed two main complaints or concerns with this type of instruction they received, one was that 'fixed pace diminished understanding' and the other was "the pressure they felt was created by the existence and form of their setted environments". The need to work at a paced fixed by the teacher was incredibly frustrating for students, particularly in the higher sets where the pace was the fastest, " I don't mind maths but when he goes ahead and you're left behind, that's when I start dreading going to maths lessons (Helen, AH, Year 10, Set 1)". When in homogeneous groupings students also felt they were constantly being compared to and judged by peers. While the basis of using homogeneous groupings is that it would motivate students and raise achievement this was found by Boaler to be true for a very small number of students.
The use of homogeneous grouping has some inherent components which are detrimental to the learning of students in the lower sets. What is at stake when we track students according to ability is perhaps the most disturbing finding of all in this chapter. Boaler refers to this as " Restricted Opportunities" students were angered by 'what they felt to be unfair restrictions on their potential mathematical achievement". Students were if anything dis-motivated to learn by tracking restrictions "they could not see the point in working in mathematics for the grades available to them" (p. 166). Students often felt that the set they were put into was not fair that the decision was based on items other than ability such as personality clashes between them and teacher or on their behaviour. Boaler uses information from Tomlinson (1987) which provided evidence that student behaviour does in fact play a role in the decision about which set they were assigned to. To me,subjectivity of the teachers feelings towards a student should not be allowed to play a part in the decision behind where that student ends up! In her summary Boaler states that 'setting' did not affect all students the same, some were indeed given an advanatage but others were negatively impacted. The consequences of being setted, especially those in lower sets was important to students and was discussed in greater detail later in the chapter.
When Boaler looked at the mixed -ability groupings at Phoenix Park and compared them to Amber Hill she found that AH students had something to react against that the PP students did not because grouping was not an issue at the latter school. The teachers at PP chose activities that gave access to students "at multiple levels" of ability, this is of course related to the newest train driving education in Newfoundland and Labrador ( and elsewhere) otherwise known as Differentiated Instruction(DI). I have to admit I don't have much experience with DI , I attended two workshops on this topic both of which were way above my level of ability as they were designed for teachers who had been working with this system and had experience using it components. These experiences left me feeling frustrated and overwhelmed and so I just used what I could glean from them in my own fashion. After reading this chapter, I am once again reminded how useful DI can be and will probably delve into a little more being careful to begin at a place where my needs are met. Obviously students at AH who, in a set , are all being taught the same material at the same pace are not having their needs met. When Boaler compared the achievement levels between AH and PP students her research found that mixed ability grouping increased achievement opportunities for all students, particularly girls and low socioeconomic status students, which lends itself to the issue of equity in mathematics.
I was profoundly disturbed by Boalers revelation that "students are rarely told the implications of the group withing which they are placed." This is shocking to me, if the knowledge of the group they are placed in must be kept secret because it is detrimental to motivation why wasn't that a flag that perhaps a closer look at this system was in order. The short term and long term effects of tracking are enormous. In the short term students go through school being compared or comparing themselves to others in an unfavorable light. Students who work at a lower level and don't know it and who are achieving well are shocked ( and understandably so) that they haven't achieved what they thought. In the long term, where you are placed at 13 determines for many the outcome of their professional or adult lives! Imagine thinking you were doing really well in math only to find out that you weren't really learning what was needed to give you the greatest options for post-secondary education, yikes! I can't believe this has been allowed. This was yet another eye-opening chapter and one I am glad to have met.
Monday, November 23, 2009
So... did I miss anything?
Just kidding!Girls, Boys and Learning Styles
Let me start by saying I can 't believe how out of touch I was with this issue (gender differences/inequity), I hadn't heard much about it of late and thought we'd moved past this, (if I thought I about it at all that is.) boy was I wrong. This was such an interesting chapter, I couldn't help but think about the whole "Men are from Mars, and Women are from Venus" thing. The differences in gender are striking in many ways! However, it is so important to come to understand why this is so in mathematics, understanding it can help us change it. I can't personally see how males are able to put aside understanding for the sake of what amounts to competitiveness" relative performance", but then they are from Mars after all!
I was glad to read that "In England, girls now attain the same proportion of the top grades in the GCSE ... as boys,and stereotyped attitudes about the irrelevance of mathematics for girls are largely disappearing." (p.137-138). The news about the top 5% of highest attaining students being largely male wasn't as encouraging, and was frankly sort of disappointing. Women valuing "connected knowing characterized by intuition,creativity and experience" is meaningful, I think. Still females do not rise to the highest levels of achievement in the same numbers as males,one has to ask what is about mathematics that causes this? Becker (1995) makes the suggestion that "girls have traditionally been denied access to success in mathematics because they tend to be connected knowers and traditional models of mathematics teaching have encouraged separate ways of working." this resonates with me, I believe Becker may be on to something here. Girls certainly seem to be at a disadvantage.
Take Amber Hill girls for example, they having been taught with the traditional approach to mathematics instruction and Boaler found that 11 out of 15 of the underachievers were females. With boys being more comfortable, playing the game, willing "..to abandon their desire for understanding and race through questions at a high speed", those (boys) from Amber Hill are at less of a disadvantage then the girls due to the differences identified. In a school where students of mathematics are operating at a disadvantage due to instructional style and speed of instruction this added layer is patently unfair to the girls.
It is disconcerting to read that the Amber Hill girls achieved less than cohorts from Phoenix Park. Think about your daughter going to this school once having been a high achiever in math and watch as her grades slip. Boaler indicates that this was the case at Amber Hill, in the group of 11 of 15 underachievers. She specifically points out that two of the girls "Carly and Lorna" attained the highest and second highest grades on the NFER respectively; then they both came out at the lowest in GCSE! To add insult to injury the 'attribution theory" findings and suggestions tend to "blame the victim". Thankfully Boaler makes the point that the girls themselves attribute their difficulties in mathematics with outside forces and not with themselves, belying attribution theory claims. Still I believe Boaler is correct when she says that "it is important not to lay the blame for their [girls] disaffection on mathematics per se because the fault lies not with the intrinsic nature of mathematics, but with school mathematics as it is commonly constructed" Gender equity in mathematics is certainly a critical issue and one that I will be paying closer attention to in the future!
Monday, November 16, 2009
Connections
I have been thinking about the Amber Hill students a lot lately, in relation to my own teaching. I believed I had made significant changes to my instructional methodology but, since starting this course, a sneaky little voice inside my head ( that won't go away now no matter how much I try to ignore it) has been telling me differently. The connection between, the lack of true understanding exhibited by the students of Amber Hill, the style of teaching by AH teachers and my own practices has been in the fore front of my thoughts. The fact that I have introduced journals and portfolios, that I use a 'problem solving' approach for some lessons and to introduce concepts do not mean I am giving my students any greater understanding then the students of AH received. This has to stop! The realization came when I read some of what was contained in the"teachingmathematics4understanding' blog. This teacher has really got it going on. She has made a sweeping change and is in my opinion doing it the right way.
I don't know if my standard of "make changes at the rate I'm most comfortable with" stance is going to cut it for much longer. That said, the idea of making wholesale change in instructional style in the middle of a year, while doing a graduate course is daunting to me. Can I wait until this summer to plan in advance for how I want to begin? The answer is, I really shouldn't, but (and its a big one) can I do justice to the kind of changes needed at this point in time? I will continue to look for ways to make the mathematics in my classroom more inquiry based. I have already been doing that. Without Jo Boaler's work, this course and our discussions I would not have been able to see that I need to do more, and I will!
I have to add that the experiences in the classroom when the students are involved in discovery and exploration of mathematics are extremely fulfilling. I have never had such enthusiasm displayed for geometry before! I do know that I need to have greater knowledge of not only the big ideas behind the mathematics myself but greater ability to analyze student thinking. My inquiry project on professional development has become an exploration of what is out there and what I need. I am now feeling as though I must take over control of my own PD and find ways to support my own learning instead of passively waiting for whatever the next session offered by district or department is. The quality of the PD sessions delivered by the district are not in question, it is just that I have needs that are not being met and I am responsible for myself. Learn Sharon Learn!
I don't know if my standard of "make changes at the rate I'm most comfortable with" stance is going to cut it for much longer. That said, the idea of making wholesale change in instructional style in the middle of a year, while doing a graduate course is daunting to me. Can I wait until this summer to plan in advance for how I want to begin? The answer is, I really shouldn't, but (and its a big one) can I do justice to the kind of changes needed at this point in time? I will continue to look for ways to make the mathematics in my classroom more inquiry based. I have already been doing that. Without Jo Boaler's work, this course and our discussions I would not have been able to see that I need to do more, and I will!
I have to add that the experiences in the classroom when the students are involved in discovery and exploration of mathematics are extremely fulfilling. I have never had such enthusiasm displayed for geometry before! I do know that I need to have greater knowledge of not only the big ideas behind the mathematics myself but greater ability to analyze student thinking. My inquiry project on professional development has become an exploration of what is out there and what I need. I am now feeling as though I must take over control of my own PD and find ways to support my own learning instead of passively waiting for whatever the next session offered by district or department is. The quality of the PD sessions delivered by the district are not in question, it is just that I have needs that are not being met and I am responsible for myself. Learn Sharon Learn!
Sunday, November 8, 2009
Interesting considering our upcoming discussion on chapter 9.
As I was searching for something else I came across this piece. I'm not even sure if it is a reliable statistic but it is interesting to me that someone could make the claim that gender issue in mathematics is gone. Below find link to the article that this visual came from.
http://www.epi.org/economic_snapshots/entry/webfeatures_snapshots_20080820/
Friday, November 6, 2009
Knowledge, Beliefs, and Mathematical Identities
I can't say I was surprised by the findings outlined in this chapter, it was already apparent where the true learning was taking place. I did enjoy, as usual, our discussion of the chapter, it was a well organized presentation for sure, the flow chart was exactly the right touch! It was what I had attempted to do in my own presentation but I couldn't get to work out the way I wanted, great job!
The effect that teachers have on students mathematical beliefs and knowledge is in some ways a little scary. In their attempt to simplify content the AH teachers caused unintentional harm to their students abilities, they could not solve problems without cues and could not apply knowledge to novel situations which required them to make connections between concepts. This is what we don't 'see' as we teach, since the development of mathematical beliefs and identities are ongoing and cumulative. What we do take as student learning, when we say, "yes they've got this" ( like Michelle expressed with her story about integers and bedmas) is generally not true at all. The more I read and hear in this course the more I realize that what we see in the classroom is that some (but not all) students can follow what I've taught and replicate it at that moment in time, but that this is not knowledge and it is definitely not understanding. We have all expressed experience with the "what is the matter with them [students] they knew this yesterday (last week, two weeks ago). Yet, like the AH teachers we didn't stop to really ask why or to connect lack of success with our teaching, "I understand it, my steps were clear... so they should understand it".
We need to strive for more than what we have accomplished in the past with our instruction. I know that having had the discussions this course work and this group has allowed I now find it impossible to ignore what I know has to be changed. I say ignore because at some level isn't that what we've all done when we notice student frustration or lack of recall, ignore the real reason behind it? What we want is to produce students who are "flexible with mathematics knowledge and are able to adapt and change", we want them to take a stance of inquiry that makes them see that exploring mathematics and connecting concepts is a natural process.
Now how do we do that exactly?
The effect that teachers have on students mathematical beliefs and knowledge is in some ways a little scary. In their attempt to simplify content the AH teachers caused unintentional harm to their students abilities, they could not solve problems without cues and could not apply knowledge to novel situations which required them to make connections between concepts. This is what we don't 'see' as we teach, since the development of mathematical beliefs and identities are ongoing and cumulative. What we do take as student learning, when we say, "yes they've got this" ( like Michelle expressed with her story about integers and bedmas) is generally not true at all. The more I read and hear in this course the more I realize that what we see in the classroom is that some (but not all) students can follow what I've taught and replicate it at that moment in time, but that this is not knowledge and it is definitely not understanding. We have all expressed experience with the "what is the matter with them [students] they knew this yesterday (last week, two weeks ago). Yet, like the AH teachers we didn't stop to really ask why or to connect lack of success with our teaching, "I understand it, my steps were clear... so they should understand it".
We need to strive for more than what we have accomplished in the past with our instruction. I know that having had the discussions this course work and this group has allowed I now find it impossible to ignore what I know has to be changed. I say ignore because at some level isn't that what we've all done when we notice student frustration or lack of recall, ignore the real reason behind it? What we want is to produce students who are "flexible with mathematics knowledge and are able to adapt and change", we want them to take a stance of inquiry that makes them see that exploring mathematics and connecting concepts is a natural process.
Now how do we do that exactly?
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