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?
Sunday, November 1, 2009
Chapter 7 - Exploring the Differences
Krista, congratulations on a job well done! Great discussion provoking questions, they made me even more sure that I want to explore how Professional Development could be used to improve instruction.
Experiencing School Mathematics has taken us through introductions and explorations into the two schools (Amber Hill and Phoenix Park) and into discussions in detail of the results of assessments for both. Now in Chapter 7, Boaler lets us hear the voices of the students as we 'Explore the Differences'. Through a mix of analysis of assessment results and interviews with the students involved we are able to see the differences in the capabilities of the students in these schools.
This chapter brings together all we have learned about the differences in curriculum and teaching styles of AH and PP. It is the proverbial 'the proof is in the pudding' scenario. Amber Hill students were able to use math knowledge, in class, when questions contained cues which indicated which 'maths' to use and when questions (GCSE) were of the short answer type. However, these same students were unable to determine for themselves which math concept/rule/procedure was needed to find a solution unless directions were explicit. When Boaler says that the 'math competencies displayed in different situations reflected both their understanding of mathematics and the belief that the students had developed about mathematics" she is discussing the student mathematical capabilities and it is clearly evident that AH students had very few.
On the bottom of page 106 I was struck by the student comments as they reveal their own realizations of how difficult they found the exam questions. It was their first inkling that there was a problem with their understanding of mathematics. Although naturally they blamed the test questions and labeled them unfair. "It's stupid really ' cause when you're in the lesson,.... you get the odd one wrong... you think well when I go into the exam I'm gonna get most of the right,'cause you get all your chapters right, But you don't. ( Alan) To me as an observer Alan's comments are a poignant reminder of the insecurity and stress that students who have little understanding feel as they are being pushed through one grade/curriculum after another. My heart really did sink as I clearly remembered being in an exam and realizing I just didn't know how to proceed, it was not good for the self esteem.
Chapter 7 gives example after example of how damaging the traditional style of teaching is for students who participate in it. They are unable to make connections between math concepts and unable to connect math to the real world. All in all, I couldn't help feeling a little sad at what educators have done to generations of students. How can we stop this from continuing? Krista asked where do we start, I think we have to start with teachers own mathematical understanding, beliefs, knowledge and attitudes. I know this for sure, we can't continue to ignore what is evident and cling to our old way of making sense of mathematics, because it just promotes a false belief that understanding has been reached when in fact nothing could be farther from the truth.
Experiencing School Mathematics has taken us through introductions and explorations into the two schools (Amber Hill and Phoenix Park) and into discussions in detail of the results of assessments for both. Now in Chapter 7, Boaler lets us hear the voices of the students as we 'Explore the Differences'. Through a mix of analysis of assessment results and interviews with the students involved we are able to see the differences in the capabilities of the students in these schools.
This chapter brings together all we have learned about the differences in curriculum and teaching styles of AH and PP. It is the proverbial 'the proof is in the pudding' scenario. Amber Hill students were able to use math knowledge, in class, when questions contained cues which indicated which 'maths' to use and when questions (GCSE) were of the short answer type. However, these same students were unable to determine for themselves which math concept/rule/procedure was needed to find a solution unless directions were explicit. When Boaler says that the 'math competencies displayed in different situations reflected both their understanding of mathematics and the belief that the students had developed about mathematics" she is discussing the student mathematical capabilities and it is clearly evident that AH students had very few.
On the bottom of page 106 I was struck by the student comments as they reveal their own realizations of how difficult they found the exam questions. It was their first inkling that there was a problem with their understanding of mathematics. Although naturally they blamed the test questions and labeled them unfair. "It's stupid really ' cause when you're in the lesson,.... you get the odd one wrong... you think well when I go into the exam I'm gonna get most of the right,'cause you get all your chapters right, But you don't. ( Alan) To me as an observer Alan's comments are a poignant reminder of the insecurity and stress that students who have little understanding feel as they are being pushed through one grade/curriculum after another. My heart really did sink as I clearly remembered being in an exam and realizing I just didn't know how to proceed, it was not good for the self esteem.
Chapter 7 gives example after example of how damaging the traditional style of teaching is for students who participate in it. They are unable to make connections between math concepts and unable to connect math to the real world. All in all, I couldn't help feeling a little sad at what educators have done to generations of students. How can we stop this from continuing? Krista asked where do we start, I think we have to start with teachers own mathematical understanding, beliefs, knowledge and attitudes. I know this for sure, we can't continue to ignore what is evident and cling to our old way of making sense of mathematics, because it just promotes a false belief that understanding has been reached when in fact nothing could be farther from the truth.
Saturday, October 24, 2009
Math Musings 2
What is effective professional development? Why this topic of inquiry? I, like other classmates, had some difficulty settling on a topic. I made a list of the issues in mathematics teaching that were critical to me. As I looked at the list trying to prioritize and choose what was most important I realized that what I was really trying to get at was how can we help teachers change, to engage, to use new methods and approaches?
To teach mathematics for understanding is not an easy undertaking if one wants to be truly effective. There are many hurdles to clear when you are changing your teaching approach. Teachers who continue to rely on direct instruction may feel they are teaching the curriculum outcomes but they are not focusing on how students will make sense of what they are trying to teach.
Some of the barriers that come between a teacher and their willingness to change include ( but are not limited too: time, understanding, beliefs, knowledge, dispositions and work load.
-teachers already spend countless unpaid overtime hours, planning, correcting, assessing, committee work etc.
-Teachers themselves are struggling with understanding the concepts they are to teach.
- the changes to assessment standards are huge
In fact, they are being bombarded with so many new issues that have identified as necessary components of teaching mathematics for understanding they become overwhelmed and shut down.
Experience and research has led me to believe that mathematics must be taught using inquiry-based methods. I have learned that teachers who are interested in making the change from traditional teaching approaches to the reform are required to obtain new pedagogical, mathematical, and professional knowledge.
At its root change is based in belief, a teacher's beliefs about math are what will influence her/his instructional decisions. To change beliefs requires a lot of work and most teachers are working hard as it is. For me its as author Maya Angelou said "When you know better you do better". For teachers to know better they must be engaged as learners. They must be helped and supported and if the support for change is not given, we will continue to see a stubborn resistance to real change in the mathematics instruction currently used in many Newfoundland and Labrador classrooms.
This is where professional development is needed. However, the usual professional development leaves something to be desired in terms of affecting change. According to Ball and Cohen, (1999) "research indicates that professional development sessions are often "intellectually superficial, disconnected from deep issues of curriculum and learning, fragmented and non-cumulative". Ball and Cohen also say that PD sessions provide little opportunity for teachers to develop deep, flexible, conceptual understanding of mathematics. So I am off on my journey to find out what makes effective professional development and the surrounding issues that make this so difficult to receive.
To teach mathematics for understanding is not an easy undertaking if one wants to be truly effective. There are many hurdles to clear when you are changing your teaching approach. Teachers who continue to rely on direct instruction may feel they are teaching the curriculum outcomes but they are not focusing on how students will make sense of what they are trying to teach.
Some of the barriers that come between a teacher and their willingness to change include ( but are not limited too: time, understanding, beliefs, knowledge, dispositions and work load.
-teachers already spend countless unpaid overtime hours, planning, correcting, assessing, committee work etc.
-Teachers themselves are struggling with understanding the concepts they are to teach.
- the changes to assessment standards are huge
In fact, they are being bombarded with so many new issues that have identified as necessary components of teaching mathematics for understanding they become overwhelmed and shut down.
Experience and research has led me to believe that mathematics must be taught using inquiry-based methods. I have learned that teachers who are interested in making the change from traditional teaching approaches to the reform are required to obtain new pedagogical, mathematical, and professional knowledge.
At its root change is based in belief, a teacher's beliefs about math are what will influence her/his instructional decisions. To change beliefs requires a lot of work and most teachers are working hard as it is. For me its as author Maya Angelou said "When you know better you do better". For teachers to know better they must be engaged as learners. They must be helped and supported and if the support for change is not given, we will continue to see a stubborn resistance to real change in the mathematics instruction currently used in many Newfoundland and Labrador classrooms.
This is where professional development is needed. However, the usual professional development leaves something to be desired in terms of affecting change. According to Ball and Cohen, (1999) "research indicates that professional development sessions are often "intellectually superficial, disconnected from deep issues of curriculum and learning, fragmented and non-cumulative". Ball and Cohen also say that PD sessions provide little opportunity for teachers to develop deep, flexible, conceptual understanding of mathematics. So I am off on my journey to find out what makes effective professional development and the surrounding issues that make this so difficult to receive.
Math Musings
As I read through articles I hope will be helpful with my inquiry project I am often struck by what I find. Recently I read "Improving Mathematics Instruction through Classroom-Based Inquiry", Ebby,Ottinger and Silver, (Teaching Children Mathematics, October 2007) Some of what I found as I read, made me think about Phoenix Park and the concerns I have already expressed in previous discussions (blog and class). To understand what I mean I think I should explain the basics of what I read.
The article describes "a university mathematics educator's efforts to support teachers in adopting a stance of critique and inquiry by developing a teacher research community" Ebby designed a course that would give teachers the opportunity to work together, research ideas and make their classrooms served as the site for inquiry into their own teaching practices.In one of the examples discussed the teacher involved discovered she was thinking wrongly about equity. She thought equity was all about allowing students total choice in all things. She based her thinking on what she had learned from Making Sense: Teaching and Learning Mathematics with Understanding ( Hiebert et al. 1997)which said that 'equity entails the assumption that all children can learn mathematics, as well as the assumption that each student must have the opportunity to learn mathematics with understanding" The teacher took this to mean that she must allow her students the choice of working together or alone and so took a hands off approach in her first cycle of inquiry. Over time she found that the students needed more 'explicit guidance' about how to work collaboratively and communicate with one another.
This is a point I raised in the last class, we expect students to work together in partners or groups and explore a problem without having shown them how. Some students may exhibit these skills naturally as part of their innate inquisitive nature, for others, the quality of learning could only be enhanced by knowledge of how communicate thinking with one another. It is true that I may not yet know everything about the preparation process the Phoenix Park teachers went through in developing the curriculum and tasks for students. I wonder if they too thought incorrectly about equity. Their hands off approach to the on or off task behaviours of students may indicate they thought students needed to be given total power of choice. Would direct guidance on how to communicate and work with these open-ended projects have benefited those students who didn't engage in this type of learning. Would their collaboration skills improved to the point where the students themselves came to value the usefulness of working this way? It will be interesting to find out.
The article describes "a university mathematics educator's efforts to support teachers in adopting a stance of critique and inquiry by developing a teacher research community" Ebby designed a course that would give teachers the opportunity to work together, research ideas and make their classrooms served as the site for inquiry into their own teaching practices.In one of the examples discussed the teacher involved discovered she was thinking wrongly about equity. She thought equity was all about allowing students total choice in all things. She based her thinking on what she had learned from Making Sense: Teaching and Learning Mathematics with Understanding ( Hiebert et al. 1997)which said that 'equity entails the assumption that all children can learn mathematics, as well as the assumption that each student must have the opportunity to learn mathematics with understanding" The teacher took this to mean that she must allow her students the choice of working together or alone and so took a hands off approach in her first cycle of inquiry. Over time she found that the students needed more 'explicit guidance' about how to work collaboratively and communicate with one another.
This is a point I raised in the last class, we expect students to work together in partners or groups and explore a problem without having shown them how. Some students may exhibit these skills naturally as part of their innate inquisitive nature, for others, the quality of learning could only be enhanced by knowledge of how communicate thinking with one another. It is true that I may not yet know everything about the preparation process the Phoenix Park teachers went through in developing the curriculum and tasks for students. I wonder if they too thought incorrectly about equity. Their hands off approach to the on or off task behaviours of students may indicate they thought students needed to be given total power of choice. Would direct guidance on how to communicate and work with these open-ended projects have benefited those students who didn't engage in this type of learning. Would their collaboration skills improved to the point where the students themselves came to value the usefulness of working this way? It will be interesting to find out.
Chapter 6 - Finding Out What They Could Do.
Scott( sorry for name mix up), you certainly had an interesting chapter to deal with. The findings that you presented helped bring together issues raised by the previous two chapters. You did a good job of highlighting the statistics in this chapter and helping me make sense of what Boaler found. Great Job, enjoyed it a lot!
As I said above, I found this chapter so very interesting. The fundamental differences between AH and PP that were exposed by Boaler certainly helped delineate what is good and what is not in mathematics education. It was helpful to me as a questioner of " what does this looks like?"(inquiry based learning, as was raised by Terri-Lynn last Thursday. Boaler's activities were designed to "require students to combine and use different areas of mathematics together", and the activities did. Students were not always successful in completing these activities and consequently showed they couldn't combine math areas because the understanding of those areas and their connections to one another had not been made/learned.
When students made 'nonsensical answers' such as a roof's angle being 200 degrees. They demonstrated what I call 'non-attending thinking'. It is obvious in this book and my own experience that students usually do not stop to ask ... does my answer make sense? More than that though they don't have that inner circuitry, that instinctual sense about math that would even cause them to pause or to think they need ask questions. Questions don't pop up because in their minds, math is not about thinking it is about doing.
Another point of interest for me were the differences in the results for year 9 students compared to year 8. There has to be a connection between the improvement and the length of time the students at PP had been involved in this kind of learning. Prior to year 8 their experiences in mathematics classrooms were essentially the same as the students in Amber Hill. More proof, to my mind at least, that inquiry or project-based learning works!
As I stated in class, although perhaps not very coherently, I have some concerns about Phoenix Parks approach. I want to make it clear that I think Phoenix Parks methods to be far superior to Amber Hills. My concern is not about the curriculum and teaching methods. It is regarding the loopholes I see in the programs structure or framework. Specifically, I am speaking about the lack of teacher redirection when students are completely off task and with the lack of organization of student written work. I am questioning whether or it would be beneficial to have some standard for student attending to tasks in order to promote their involvement with the mathematics at hand. Further to that thought, I wonder if the PP teachers can even determine if there are areas of content that have not been covered at all. I do understand the need to build an atmosphere of openness and trust so that students will continue to explore, inquire, question, and contend with the solution to a problem. As for organization of written record of work, its as Dr. Stordy pointed the other night,assessment is not just about what the students put down on paper. Still I wonder how one could sort through the mess of papers to find out what the students recorded and what if anything the written record says about that students needs. I'm sure the answers to some of this will become clearer as we progress through the book. Obviously something informs the project formation by teachers, it is possible my questions come from my limited experience with these methods and that these concerns are really not valid at all. We'll see!
As I said above, I found this chapter so very interesting. The fundamental differences between AH and PP that were exposed by Boaler certainly helped delineate what is good and what is not in mathematics education. It was helpful to me as a questioner of " what does this looks like?"(inquiry based learning, as was raised by Terri-Lynn last Thursday. Boaler's activities were designed to "require students to combine and use different areas of mathematics together", and the activities did. Students were not always successful in completing these activities and consequently showed they couldn't combine math areas because the understanding of those areas and their connections to one another had not been made/learned.
When students made 'nonsensical answers' such as a roof's angle being 200 degrees. They demonstrated what I call 'non-attending thinking'. It is obvious in this book and my own experience that students usually do not stop to ask ... does my answer make sense? More than that though they don't have that inner circuitry, that instinctual sense about math that would even cause them to pause or to think they need ask questions. Questions don't pop up because in their minds, math is not about thinking it is about doing.
Another point of interest for me were the differences in the results for year 9 students compared to year 8. There has to be a connection between the improvement and the length of time the students at PP had been involved in this kind of learning. Prior to year 8 their experiences in mathematics classrooms were essentially the same as the students in Amber Hill. More proof, to my mind at least, that inquiry or project-based learning works!
As I stated in class, although perhaps not very coherently, I have some concerns about Phoenix Parks approach. I want to make it clear that I think Phoenix Parks methods to be far superior to Amber Hills. My concern is not about the curriculum and teaching methods. It is regarding the loopholes I see in the programs structure or framework. Specifically, I am speaking about the lack of teacher redirection when students are completely off task and with the lack of organization of student written work. I am questioning whether or it would be beneficial to have some standard for student attending to tasks in order to promote their involvement with the mathematics at hand. Further to that thought, I wonder if the PP teachers can even determine if there are areas of content that have not been covered at all. I do understand the need to build an atmosphere of openness and trust so that students will continue to explore, inquire, question, and contend with the solution to a problem. As for organization of written record of work, its as Dr. Stordy pointed the other night,assessment is not just about what the students put down on paper. Still I wonder how one could sort through the mess of papers to find out what the students recorded and what if anything the written record says about that students needs. I'm sure the answers to some of this will become clearer as we progress through the book. Obviously something informs the project formation by teachers, it is possible my questions come from my limited experience with these methods and that these concerns are really not valid at all. We'll see!
Michelle 's Presentation
I just realized that in my blog on chapter 5 I hadn't extended congratulations to Michelle on a job well done! Your presentation was wonderful, it was clear and concise it helped us track and make sense of what was said in the chapter. Good on you... as the British would say! ... or is that Australian? , never mind, still applies. Sorry for being tardy with comments!
Sunday, October 18, 2009
Phoenix Park
Standard 1 of NCTM's 'Professional Standards',(1991)is entitled "Worthwhile Mathematics Tasks"
The teacher of mathematics should pose tasks that are based on-sound and significant mathematics:
-knowledge of students' understandings, interests, and experiences;
-knowledge of the range of ways that diverse students learn mathematics
and that:
-engage students' intellect;
-develop students' mathematical understandings and skills;
-stimulate students to make connections and develop a coherent framework for
mathematical ideas;
-call for problem formulation, problem solving, and mathematical reasoning;
- promote communication about mathematics;
- represent mathematics as an ongoing human activity;
-display sensitivity to, and draw on, students' diverse background experiences
and dispositions;
and
-promote the development of all students' dispositions to do mathematics.
Phoenix Park's mathematics program certainly met this standard. The curriculum at Phoenix Park was teacher designed, it required the teachers to "know a lot about the students - what they knew what would be most helpful for them to work on" in this they meet the descriptors from the NCTM standards. In fact the mathematics tasks that the students of Phoenix Park were given to work on allowed for openness and creativity. The teachers supported this creativity by making "deliberate efforts not to structure the work for students", they did not give closed answers to student questions instead they would reform the question in a way that invited the students to explain what they knew and to identify what they needed to find out, something I am currently working on in my own instructional style. The students were guided to make connections and to reason and communicate their thinking and in doing so developed a mathematics disposition that was based on the belief that it was more important to think in mathematics than to remember rules. I was very impressed with the learning opportunities for students at Phoenix Park.
It was very surprising to me that time on task for both Amber Hill and Phoenix Park was about equal. I would say that I had a very strong reaction to the descriptions of students being permitted to wander at will and to be noticeably off task for long periods. There was no attempt by the teacher to refocus or encourage them to return to the work at hand and I found this disturbing. I couldn't help but think that a little structure in this area would have benefited the students. I can only suppose that the teachers felt it more important to create an atmosphere in the classroom where students were unafraid of being wrong and willing to explore mathematics concepts then it was to impose any useful level of discipline. I'm not 100% sure that they aren't right in this but I'm a long way from this in my teaching.
The teacher of mathematics should pose tasks that are based on-sound and significant mathematics:
-knowledge of students' understandings, interests, and experiences;
-knowledge of the range of ways that diverse students learn mathematics
and that:
-engage students' intellect;
-develop students' mathematical understandings and skills;
-stimulate students to make connections and develop a coherent framework for
mathematical ideas;
-call for problem formulation, problem solving, and mathematical reasoning;
- promote communication about mathematics;
- represent mathematics as an ongoing human activity;
-display sensitivity to, and draw on, students' diverse background experiences
and dispositions;
and
-promote the development of all students' dispositions to do mathematics.
Phoenix Park's mathematics program certainly met this standard. The curriculum at Phoenix Park was teacher designed, it required the teachers to "know a lot about the students - what they knew what would be most helpful for them to work on" in this they meet the descriptors from the NCTM standards. In fact the mathematics tasks that the students of Phoenix Park were given to work on allowed for openness and creativity. The teachers supported this creativity by making "deliberate efforts not to structure the work for students", they did not give closed answers to student questions instead they would reform the question in a way that invited the students to explain what they knew and to identify what they needed to find out, something I am currently working on in my own instructional style. The students were guided to make connections and to reason and communicate their thinking and in doing so developed a mathematics disposition that was based on the belief that it was more important to think in mathematics than to remember rules. I was very impressed with the learning opportunities for students at Phoenix Park.
It was very surprising to me that time on task for both Amber Hill and Phoenix Park was about equal. I would say that I had a very strong reaction to the descriptions of students being permitted to wander at will and to be noticeably off task for long periods. There was no attempt by the teacher to refocus or encourage them to return to the work at hand and I found this disturbing. I couldn't help but think that a little structure in this area would have benefited the students. I can only suppose that the teachers felt it more important to create an atmosphere in the classroom where students were unafraid of being wrong and willing to explore mathematics concepts then it was to impose any useful level of discipline. I'm not 100% sure that they aren't right in this but I'm a long way from this in my teaching.
Wednesday, October 14, 2009
Chapter 4 follow up
As I read through chapter 5, my thoughts returned time and again to Chapter 4. I felt like I had a little more to say I guess and wanted to make this post before we discuss the next chapter. The more I read the more I realize how much there is to do to make a classroom an inviting, exciting arena for mathematical learning/understanding. The teachers at Amber Hill, however good intentioned they may be, were not being effective, shouldn't they have been able to see that? I turn the mirror on myself and ask if I have not also been guilty of this very thing. I pay much closer attention now to the what and why when planning and get a great sense of satisfaction from knowing that the small changes I've made can be built upon and its all good for the students!
As long as I'm looking in that mirror I must admit that I have been guilty of 'excessive prompting' in the past. Allowing students to think, that break between question/problem and answer is excruciating. I have noted that the students have noticed the changes too, at times they have such quizzical looks on their faces as in " When is Miss going to step in?" It is a bit of a tight-rope walk and a lot of getting rid of bad habits to know when is the right time to step in. I struggle on! What has inspired me to be aware is the idea that Boaler presented from her observations at Amber Hill " The teachers thought that students would not or could not think" this " learned helplessness" is something I do not want to be a part of continuing. I know there will be times when I will 'fall of the wagon' so to speak but it has become an important issue for me.
Finally, when I think about all that I read and learned in chapter 4, differences between teacher beliefs and actions, the development of negative student attitudes toward mathematics, the lack of true mathematical understanding have all been eye opening for me. However, it is the idea that the students from Amber Hill were considered to have inadequacies based on the social group they were identified as having come from, that bothered me the most. The studies Anyon, ( 1981) cites that find that schools in lower socio-economic areas "discouraged personal assertiveness and intellectual inquisitiveness in students and assigned work that most often involved substantial amounts of rote activity" is perhaps, shameful! ( maybe that's too strong a word? Hmm?) As I comb through research for this course I have come across the following quote, "All students regardless of their personal characteristics, backgrounds, or physical characteristics must have the opportunity to study - and support to learn - mathematics (NCTM, 2000, p.12) hallelujah brother, say it again! Equity is certainly one of the critical issues facing mathematics education today, Amber Hill certainly demonstrates that.
As long as I'm looking in that mirror I must admit that I have been guilty of 'excessive prompting' in the past. Allowing students to think, that break between question/problem and answer is excruciating. I have noted that the students have noticed the changes too, at times they have such quizzical looks on their faces as in " When is Miss going to step in?" It is a bit of a tight-rope walk and a lot of getting rid of bad habits to know when is the right time to step in. I struggle on! What has inspired me to be aware is the idea that Boaler presented from her observations at Amber Hill " The teachers thought that students would not or could not think" this " learned helplessness" is something I do not want to be a part of continuing. I know there will be times when I will 'fall of the wagon' so to speak but it has become an important issue for me.
Finally, when I think about all that I read and learned in chapter 4, differences between teacher beliefs and actions, the development of negative student attitudes toward mathematics, the lack of true mathematical understanding have all been eye opening for me. However, it is the idea that the students from Amber Hill were considered to have inadequacies based on the social group they were identified as having come from, that bothered me the most. The studies Anyon, ( 1981) cites that find that schools in lower socio-economic areas "discouraged personal assertiveness and intellectual inquisitiveness in students and assigned work that most often involved substantial amounts of rote activity" is perhaps, shameful! ( maybe that's too strong a word? Hmm?) As I comb through research for this course I have come across the following quote, "All students regardless of their personal characteristics, backgrounds, or physical characteristics must have the opportunity to study - and support to learn - mathematics (NCTM, 2000, p.12) hallelujah brother, say it again! Equity is certainly one of the critical issues facing mathematics education today, Amber Hill certainly demonstrates that.
Saturday, October 10, 2009
What's Current, What's useful, What's useable?
Inquiry project research has led me down some very different ( and interesting) roads, I'm not the most adept researcher. Just lately I've been considering changing my topic, but haven't reached a decision. One of the reasons is the difficulty I'm having finding information on original topic but mainly its because I've become interested in a couple of other ideas that have cropped up.
New idea maybe: Self -regulated Mathematics Learning??
This area of thought is intriguing to me and seems destined to be connected to differentiated instruction ( althought i've yet to find the connection). Perhaps choosing this will allow me to 'kill two birds with one stone' as I am on the starting end of the learning curve about DI. Mainly I'm interested in any concept or approach that can help inquiry-based learning of mathematics. Students who are good 'problem-solvers' need, I believe, a strong background in mathematics knowledge ( facts,symbols, definitions, algorithms etc) but it is not necessary for this 'knowledge' to be received only through direct-instruction. I truly believe that if we make teachers aware of the 'how to' and not just the 'why' of teaching using an inquiry-based approach we will see it used more and more as an instructional approach. Perhaps my 'inquiry' for this course will lead me to a place that will be helpful in this regard.
I am never sure when an idea, that seems new to me, is actually still current in the field of mathematics. Is self-regulated learning a viable topic or have we moved on .... I guess that's a good question for Thursday!
New idea maybe: Self -regulated Mathematics Learning??
This area of thought is intriguing to me and seems destined to be connected to differentiated instruction ( althought i've yet to find the connection). Perhaps choosing this will allow me to 'kill two birds with one stone' as I am on the starting end of the learning curve about DI. Mainly I'm interested in any concept or approach that can help inquiry-based learning of mathematics. Students who are good 'problem-solvers' need, I believe, a strong background in mathematics knowledge ( facts,symbols, definitions, algorithms etc) but it is not necessary for this 'knowledge' to be received only through direct-instruction. I truly believe that if we make teachers aware of the 'how to' and not just the 'why' of teaching using an inquiry-based approach we will see it used more and more as an instructional approach. Perhaps my 'inquiry' for this course will lead me to a place that will be helpful in this regard.
I am never sure when an idea, that seems new to me, is actually still current in the field of mathematics. Is self-regulated learning a viable topic or have we moved on .... I guess that's a good question for Thursday!
Thursday, October 8, 2009
Critical Reflections - Chapter 4 and My Presentation
I would like to begin by thanking everyone for participating in the discussion surrounding the issues I found in chapter 4, what a relief!
Amber Hill is a school with some critical issues in mathematics that need resolving. The teachers are caring, more than competent, "All the mathematics teachers were well qualified mathematics specialists", and efficient. Their effectiveness, however, is an entirely different matter. I do believe it would be wrong to say that 'no' mathematics learning took place at all ( and Boaler doesn't) but it must also be said that very little true mathematical knowledge or understanding happened.
To read about these classrooms is to recognize the forms of instruction, the ways students felt, the motivations behind teachers' actions. The old adage " people in glass houses shouldn't throw stones" comes to mind. It was heartening to read that "students did not blame their boredom on the intrinsic nature of mathematics" Still we sure have a long way to go folks! See you next Thursday!
Amber Hill is a school with some critical issues in mathematics that need resolving. The teachers are caring, more than competent, "All the mathematics teachers were well qualified mathematics specialists", and efficient. Their effectiveness, however, is an entirely different matter. I do believe it would be wrong to say that 'no' mathematics learning took place at all ( and Boaler doesn't) but it must also be said that very little true mathematical knowledge or understanding happened.
To read about these classrooms is to recognize the forms of instruction, the ways students felt, the motivations behind teachers' actions. The old adage " people in glass houses shouldn't throw stones" comes to mind. It was heartening to read that "students did not blame their boredom on the intrinsic nature of mathematics" Still we sure have a long way to go folks! See you next Thursday!
Critical Reflections - chapters 1-3
I guess I was caught up in preparing for my presentation, I've just realized I hadn't posted a reflection on Chapters 1-3, mea culpa Luckily I'd made a few notes, here they are:
After reading Schoenfeld's introduction, it did in fact "induce me to read on",( of course the fact its required reading played a part too!) This being my first graduate level course I am unfamiliar with the rigors of research and what makes a reliable study, suffice it to say I was suitably impressed by Boaler's explanation of her study and have no doubts that her findings are accurate and founded in truth. I was struck but the inclusion (in both Schoenfeld's and Boaler's introductions) of gender as an issue. I must have been living with my head in the sand, I so thought that had been dealt with ( this was an issue yea those many years ago when I did my Bachelor's degree) . Upon reflection I wonder if it wasn't because I didn't see gender in math. In other words I don't expect the boys to do better than the girls, I just expect ( and find) that there are always a mixture of abilities in any classroom.
I was also struck by the idea of reality versus facade in education. When Boaler described Amber Hill, its Principal, its reception area, classroom/hallway behaviour, I started to get a picture of the school, as was her intention. This was then contrasted with what was happening at Amber Hill,( type of instruction, level of learning etc) and one quickly realized that 'you have bite into the chocolate to find out what flavor is on the inside'. I'm sure as we progress through the book Amber Hill will reveal itself to have some redeeming qualities. I look forward to the journey!
Postscript: Now that I've looked so closely at chapter 4 I realize that as Boaler states (p.47) "the portrayal of mathematics at Amber Hill is quite bleak" More about that in my next entry!
After reading Schoenfeld's introduction, it did in fact "induce me to read on",( of course the fact its required reading played a part too!) This being my first graduate level course I am unfamiliar with the rigors of research and what makes a reliable study, suffice it to say I was suitably impressed by Boaler's explanation of her study and have no doubts that her findings are accurate and founded in truth. I was struck but the inclusion (in both Schoenfeld's and Boaler's introductions) of gender as an issue. I must have been living with my head in the sand, I so thought that had been dealt with ( this was an issue yea those many years ago when I did my Bachelor's degree) . Upon reflection I wonder if it wasn't because I didn't see gender in math. In other words I don't expect the boys to do better than the girls, I just expect ( and find) that there are always a mixture of abilities in any classroom.
I was also struck by the idea of reality versus facade in education. When Boaler described Amber Hill, its Principal, its reception area, classroom/hallway behaviour, I started to get a picture of the school, as was her intention. This was then contrasted with what was happening at Amber Hill,( type of instruction, level of learning etc) and one quickly realized that 'you have bite into the chocolate to find out what flavor is on the inside'. I'm sure as we progress through the book Amber Hill will reveal itself to have some redeeming qualities. I look forward to the journey!
Postscript: Now that I've looked so closely at chapter 4 I realize that as Boaler states (p.47) "the portrayal of mathematics at Amber Hill is quite bleak" More about that in my next entry!
Sunday, September 27, 2009
Sir Ken Robinson
What a difference 15 minutes can make to a persons views on learning. I watched Sir Ken Robinson make a case for creating and supporting the need for creativity in education. I couldn't help but see the correlation between what Sir Ken calls "educating people out of creativity",(2005) and the difficulty students have with making connections among and between math concepts and processes . I have watched students struggle to communicate their mathematical reasoning. I have seen that many are unable to apply learned concepts in new or different circumstances. I have long questioned if the difficulties students were demonstrating were due to the fact that children may not have the cognitive maturity to isolate their thought processes as is needed in the latest approach to mathematics . Otherwise, as my thinking went, wouldn't the reform in the approach to instruction of mathematics that has been in practice for more than 10 years, have mediated this weakness and produced children who could explain their thinking?
Now it occurs to me that the root of the problems we are seeing in the learning of mathematics may be due to the fact that we have an educational system that has, as Sir Ken says, " educated people out of creativity" (2005). Sir Ken makes the argument that creativity is as important as literacy and that our education system is failing to cultivate this, (In a later speech at the Apple Educational Leadership Summit (2008) he added that creativity is as important as literacy and numeracy.) If creativity is not valued, if students learn early that mistakes are the worse things you can make how can we expect them to truly understand the underlying concepts in mathematics. Without creativity, without being willing to try new ideas and be wrong, students quickly turn to the teacher to show them the 'right' way to do math. In other words, if creativity was valued as much as literacy and numeracy then perhaps students ability to create understanding and learning in any area through investigation would be a natural process and as such more successful.
In our school district we have the following 'specialist' positions, Instruction and School Leadership , Student Support Services, Math/ Technology, Science/Technology,Primary/Elementary Math, Primary/Elementary literacy but no where is there mention of creativity. It is mind boggling to realize that the system we have invested so much in was founded on a model designed, according to Sir Ken, to meet the needs of the industrial revolution! It is ironic that the shift in thinking and instruction of math that teachers have been asked to make is based on teaching children to see 'the big ideas' in math. Seeing the big ideas means being able to think outside the box,make the connections, understand the underlying processes are all necessary in mathematics. However, the fact that even futurists are unable to predict what the world will look like in 5 years begs the question just who needs to look at the bigger picture? Sir Ken makes a very good case for changing our education system to one that nurtures creativity, not just for mathematics, but to better prepare students to have 'live lives of purpose and meaning"(Robinson, 2008). Isn't this the point of what we do after all?
Now it occurs to me that the root of the problems we are seeing in the learning of mathematics may be due to the fact that we have an educational system that has, as Sir Ken says, " educated people out of creativity" (2005). Sir Ken makes the argument that creativity is as important as literacy and that our education system is failing to cultivate this, (In a later speech at the Apple Educational Leadership Summit (2008) he added that creativity is as important as literacy and numeracy.) If creativity is not valued, if students learn early that mistakes are the worse things you can make how can we expect them to truly understand the underlying concepts in mathematics. Without creativity, without being willing to try new ideas and be wrong, students quickly turn to the teacher to show them the 'right' way to do math. In other words, if creativity was valued as much as literacy and numeracy then perhaps students ability to create understanding and learning in any area through investigation would be a natural process and as such more successful.
In our school district we have the following 'specialist' positions, Instruction and School Leadership , Student Support Services, Math/ Technology, Science/Technology,Primary/Elementary Math, Primary/Elementary literacy but no where is there mention of creativity. It is mind boggling to realize that the system we have invested so much in was founded on a model designed, according to Sir Ken, to meet the needs of the industrial revolution! It is ironic that the shift in thinking and instruction of math that teachers have been asked to make is based on teaching children to see 'the big ideas' in math. Seeing the big ideas means being able to think outside the box,make the connections, understand the underlying processes are all necessary in mathematics. However, the fact that even futurists are unable to predict what the world will look like in 5 years begs the question just who needs to look at the bigger picture? Sir Ken makes a very good case for changing our education system to one that nurtures creativity, not just for mathematics, but to better prepare students to have 'live lives of purpose and meaning"(Robinson, 2008). Isn't this the point of what we do after all?
Monday, September 21, 2009
When Good Teaching Leads to Bad Results - Schoenfeld
As a teacher of mathematics, Schoenfeld's (1988) article " When Good Teaching Leads to Bad Results: The Disasters of " Well Taught" Mathematics Courses" made several significant points which are as pertinent to Mathematics Education today as they were when the article was first written. The issues of what constitutes actual learning in mathematics and which approach (traditional or reform) provides the best instruction for that learning are, for me, ideas I have pondered as a mathematics teacher today.
I struggle with providing students experiences which cover the curriculum but also develop understanding and I know that I am not alone. Many teachers express the same sorts of frustrations with the current program. This article has helped me to see the bigger picture and has put into words what, up to now, has been a collection of vague notions about why we struggle so much with math instruction. Schoenfeld's discussion of the delineation between "becoming competent at performing the symbolic manipulation procedures... and grasping the underlying mathematical ideas..." (Schoenfeld, 1988 p.3) hits at the heart of problems we and students face with the current math program that is offered in Newfoundland.
Are we developing competency with understanding, can any one program provide students with all that they need? The 'problem solving' approach to mathematics that has been toted by our school district and our Department of Education should provide students with enough experiences that true understanding is gained. Yet, many teachers and parents believe that we are turning out students who do not have the 'basic skills' in mathematical procedures let alone 'understanding'. What has been referred by many as ' math wars'. It does not seem that we are providing what students need. How much of this is due to teachers of elementary math being 'non-specialists' in this field? The majority of us are not mathematicians in any sense of the word. We did not complete a math degree, but instead completed the minimum number of courses required for our Bachelor of Education degrees. Whatever the reason or reasons it is evident that we should at least be aware of the misconceptions about math that our instruction may be causing our students to develop.
Schoenfeld, highlighted a set of 4 'beliefs' that students in his case study had developed about math and the misconceptions these beliefs caused. I can relate to much of what he has stated because it represents my experience with math as a student and teacher. I believed, perhaps without realizing it, that "students who understand...can solve the math problem in five minutes or less". This was the very reason I gave up so easily when confronted with math work in my own school days. This is the reason I continuously tell my students that not knowing the answer right away does not mean you can't figure it out. I also always believed that only mathematicians really understood math, and if I reflect on this point I have to say that this was a belief I held right up until I read this article. Now I am questioning my thinking surrounding this idea. I am not totally convinced that there isn't a good deal of truth in "only geniuses are capable of discovering, creating or really understanding mathematics" as my statements about mathematicians vs math teachers in previous paragraphs attests to. Still this has certainly given me food for thought.
This article has highlighted for me what some of the critical issues in mathematics education are. It has provided me with a framework for many of the conclusions I have come to as a mathematics teacher. I know that many students do experience success with the math curriculum to one degree or another. I also must conclude that in my experience it is true that students see themselves as " passive consumers of others' mathematics' and that connections between concepts and solutions to problems are not always demonstrated. Making connections between process/procedure and understanding the 'big ideas' behind the concept are simply few and far between.
My own son has always received very high marks in mathematics all the way to the twelfth grade where, we hope, he will continue to do so. So as I read through Schoenfelds article I decided to ask him to complete the problem that Wetheimer ( 1959) asked elementary school children to solve. I showed him the fractions as seen on page 5 and asked him if he though he could solve this. " Yes, sure I can" was his reply. However, when I asked him how he would solve it he gave a step by step procedure that included adding and dividing the numbers involved. When I prompted him to use what he knows about operations and fractions to find another way to get the answer he couldn't procede. He was stunned when I pointed out that there was no need to calculate because the relationship between fractions and division and the fact that inverse operations cancel each other out. Then, of course, he wanted to prove himself, " Give me another one" so I did. I gave him pencil and paper and asked him " If a bus holds 36 passengers how many buses would be needed to get 1128 soldiers to training. Again he quickly divided and told me 31, you can imagine his chagrin when I asked how the remaining 12 soldier were going to get where they needed to be. I believe him to be very mathematical and yet he did not make the needed connections. I can't help but wonder that if Wertheimer's problem came from 1959 and the problem from the third National Assessment of Educational Progress (Carpenter, Lindquist, Matthews and Silver,1983). was given more than 20 years ago how much progress have we made in our approach to teaching mathematics?
I struggle with providing students experiences which cover the curriculum but also develop understanding and I know that I am not alone. Many teachers express the same sorts of frustrations with the current program. This article has helped me to see the bigger picture and has put into words what, up to now, has been a collection of vague notions about why we struggle so much with math instruction. Schoenfeld's discussion of the delineation between "becoming competent at performing the symbolic manipulation procedures... and grasping the underlying mathematical ideas..." (Schoenfeld, 1988 p.3) hits at the heart of problems we and students face with the current math program that is offered in Newfoundland.
Are we developing competency with understanding, can any one program provide students with all that they need? The 'problem solving' approach to mathematics that has been toted by our school district and our Department of Education should provide students with enough experiences that true understanding is gained. Yet, many teachers and parents believe that we are turning out students who do not have the 'basic skills' in mathematical procedures let alone 'understanding'. What has been referred by many as ' math wars'. It does not seem that we are providing what students need. How much of this is due to teachers of elementary math being 'non-specialists' in this field? The majority of us are not mathematicians in any sense of the word. We did not complete a math degree, but instead completed the minimum number of courses required for our Bachelor of Education degrees. Whatever the reason or reasons it is evident that we should at least be aware of the misconceptions about math that our instruction may be causing our students to develop.
Schoenfeld, highlighted a set of 4 'beliefs' that students in his case study had developed about math and the misconceptions these beliefs caused. I can relate to much of what he has stated because it represents my experience with math as a student and teacher. I believed, perhaps without realizing it, that "students who understand...can solve the math problem in five minutes or less". This was the very reason I gave up so easily when confronted with math work in my own school days. This is the reason I continuously tell my students that not knowing the answer right away does not mean you can't figure it out. I also always believed that only mathematicians really understood math, and if I reflect on this point I have to say that this was a belief I held right up until I read this article. Now I am questioning my thinking surrounding this idea. I am not totally convinced that there isn't a good deal of truth in "only geniuses are capable of discovering, creating or really understanding mathematics" as my statements about mathematicians vs math teachers in previous paragraphs attests to. Still this has certainly given me food for thought.
This article has highlighted for me what some of the critical issues in mathematics education are. It has provided me with a framework for many of the conclusions I have come to as a mathematics teacher. I know that many students do experience success with the math curriculum to one degree or another. I also must conclude that in my experience it is true that students see themselves as " passive consumers of others' mathematics' and that connections between concepts and solutions to problems are not always demonstrated. Making connections between process/procedure and understanding the 'big ideas' behind the concept are simply few and far between.
My own son has always received very high marks in mathematics all the way to the twelfth grade where, we hope, he will continue to do so. So as I read through Schoenfelds article I decided to ask him to complete the problem that Wetheimer ( 1959) asked elementary school children to solve. I showed him the fractions as seen on page 5 and asked him if he though he could solve this. " Yes, sure I can" was his reply. However, when I asked him how he would solve it he gave a step by step procedure that included adding and dividing the numbers involved. When I prompted him to use what he knows about operations and fractions to find another way to get the answer he couldn't procede. He was stunned when I pointed out that there was no need to calculate because the relationship between fractions and division and the fact that inverse operations cancel each other out. Then, of course, he wanted to prove himself, " Give me another one" so I did. I gave him pencil and paper and asked him " If a bus holds 36 passengers how many buses would be needed to get 1128 soldiers to training. Again he quickly divided and told me 31, you can imagine his chagrin when I asked how the remaining 12 soldier were going to get where they needed to be. I believe him to be very mathematical and yet he did not make the needed connections. I can't help but wonder that if Wertheimer's problem came from 1959 and the problem from the third National Assessment of Educational Progress (Carpenter, Lindquist, Matthews and Silver,1983). was given more than 20 years ago how much progress have we made in our approach to teaching mathematics?
Tuesday, September 15, 2009
Math Autobiography
If you could look into any of my childhood classrooms you would undoubtedly see chalkboards and erasers, over sized compasses and protractors, colored chalk and yard sticks, children in rows, note and text books open. The teacher would be stationed at the front explaining that days lessons with examples and diagrams. Work set and children expected to complete it independently if they had listened properly. My math autobiography is probably very similiar to many of the people who struggled their way through the school system of the 70's ( okay 60's and 70's). To be fair, I do not recall very many specifics of 'what math looked like', particularly in the younger grades, but those I do are not positive memories.
Which is strange because at the time of the kindergarten registration, I can clearly remember being excited about the puzzles and books and the big fat red pencils we would be using. I also remember seeing colored beads of different shapes strung together and wanting to be able to play with the items I could see sitting in an open cupboard not far from where I sat next to Mom. I don't know when that feeling of excitement about school disappeared but it didn't take long. When Sir Ken Richardson was talking about 'being educated out of creativity' on the video we watched during the first class it resonated with me.
I do remember in grade 2 being hit on knuckles ( over and over) with a ruler for getting my 'sums' wrong and being frightened and confused because it was not clear to me what I was doing wrong. Otherwise I just recall that math was something I found hard and confusing and a real challenge to understand, and then came grade 6. Grade 6 has the distinction of holding both my best and worst math memories. The best was a lesson on fractions that included a recipe for a cherry cake, which I went home and made and brought to school. It was the only time all year I felt I had pleased the teacher. She always seem to be moving too fast and seemed to only have time and patience with the smartest kids in class. I was definitely not one of those in fact I was coming to the conclusion that I must be stupid and that feeling was about to get stronger.
Pardon me while I digress for a moment buy I feel I should explain that the education system in Newfoundland at the time tracked students according to ability. The A class having the students of highest ability who would get mostly 80- 100% . B class had students who would most likely achieve in the 70-80% range I guess and I had always been in this group.
At the end of grade 6 the teacher approached me about being moved to the C group in grade 7 because of math. It was quite a blow to my self esteem and cemented in me the notion that I indeed could not do math and thus is probably the worst math memory I have. It didn't' matter that they moved me back to the B class in grade 8, it was too late I had already joined the ranks of the ' I can't do math society" a subsidiary of the " I hate math foundation".
That change in outlook lasted until my first year university when I had my epiphany regarding math. I had registered for the first level math course and was struggling ( I had been out of high school for 5 years before starting at MUN) . When I asked the professor for help she told me that if I didn't understand it I should drop the course and take the foundation math course which was noncredit. Well it was past the drop/add deadline and I couldn't afford to take this course again. Failing was not an option! So before the first exam I went to the library and booked one of those little study rooms. I tried to do the problems we had worked on in class, but always found myself getting the same wrong answer. The frustration would build and I can still see myself slamming books down and flipping pages wildly, stamping out of the room with a " I can't do this" sense of fatalism. Then I'd make myself go back and try again and a strange thing happened on the way to giving up, a light went on. I figured out what I was doing wrong. What elation, what a feeling of accomplishment what a change in my view of math! Math was I realized something I could do if I was willing to work long and hard. Not knowing the answer right away no longer meant "I can't do this" it meant that I had to keep working.
Giving up was the way I had always faced math. Math teachers in general didn't give me a lot of time, and it felt as if they seemed to agree that I was beyond help and so very little was ever given ( although to be honest this was more of subconscious feeling) . Except in grade 10 when I had a math teacher who expected me to work hard, who checked to see if my homework was done, who kept me after to class to grill me if it wasn't and who called on me to answer questions as regularly as he did anyone else. This feeling of having to live up to his standard was in equal parts frustrating and motivational. He was the best! The fact that he took his time, that he actually looked at the class when asking if anyone needed help instead of having his back to the room while erasing the board were all factors in making him stand out as a good math teacher. He loved the subject and he liked us and everyone knew it.
This could not be said about any of my other high school math teachers. Assignments usually consisted of pages of repeated practice problems. Similar to regular math homework but printed in hand on a mimeographed sheet. Assessment consisted of marks given for end of unit tests, some small weight given for homework completion, assignments were marked and of course the ubiquitous mid-term and final exams. Summative evaluation was the norm for all the years I was in school and was entrenched as the way things were done. This pattern continued into university through the 4 math courses I took while there. The courses at university consisted of the two first year entrance level courses, Primary/ Elementary Math 1 and 2 and the Math Methods course. I may have had that epiphany but I still didn't actually like math.
So it continued, I taught math in many grades as a substitute and replacement teacher and felt I was doing a good job. I did what all the other teacher seemed to be doing, what my teachers did. I taught, they listened, they set to work on problems. This was how it was until they brought in a new math program and it made its way to grade 5 several years ago. It was a totally new approach to teaching and assessing students. Assessment became multi-layered and the problem solving approach to mathematics was de rigeur. What horror, oh the complaining and gnashing of teeth in the staffroom. I struggled to make sense of it while clinging to the ways I had always taught. But the struggle became to big and I gave myself over to change. I have put in a tremendous amount of time into changing my approach to the teaching of math and I can see the difference this program has made to so many students. Having said that I must add that in my opinion no one approach can meet all the needs. Differentiation sounds like it may hold the key and is something that I have included in my professional growth plan. These changes in my attitude have developed over a good many years. I have gone from hopeless to hopeful. I feel that mathematics can be an exciting and rewarding experience for all students. While I struggle to make it so, I know that time and effort pays off and perserverence is the name of the game!
Which is strange because at the time of the kindergarten registration, I can clearly remember being excited about the puzzles and books and the big fat red pencils we would be using. I also remember seeing colored beads of different shapes strung together and wanting to be able to play with the items I could see sitting in an open cupboard not far from where I sat next to Mom. I don't know when that feeling of excitement about school disappeared but it didn't take long. When Sir Ken Richardson was talking about 'being educated out of creativity' on the video we watched during the first class it resonated with me.
I do remember in grade 2 being hit on knuckles ( over and over) with a ruler for getting my 'sums' wrong and being frightened and confused because it was not clear to me what I was doing wrong. Otherwise I just recall that math was something I found hard and confusing and a real challenge to understand, and then came grade 6. Grade 6 has the distinction of holding both my best and worst math memories. The best was a lesson on fractions that included a recipe for a cherry cake, which I went home and made and brought to school. It was the only time all year I felt I had pleased the teacher. She always seem to be moving too fast and seemed to only have time and patience with the smartest kids in class. I was definitely not one of those in fact I was coming to the conclusion that I must be stupid and that feeling was about to get stronger.
Pardon me while I digress for a moment buy I feel I should explain that the education system in Newfoundland at the time tracked students according to ability. The A class having the students of highest ability who would get mostly 80- 100% . B class had students who would most likely achieve in the 70-80% range I guess and I had always been in this group.
At the end of grade 6 the teacher approached me about being moved to the C group in grade 7 because of math. It was quite a blow to my self esteem and cemented in me the notion that I indeed could not do math and thus is probably the worst math memory I have. It didn't' matter that they moved me back to the B class in grade 8, it was too late I had already joined the ranks of the ' I can't do math society" a subsidiary of the " I hate math foundation".
That change in outlook lasted until my first year university when I had my epiphany regarding math. I had registered for the first level math course and was struggling ( I had been out of high school for 5 years before starting at MUN) . When I asked the professor for help she told me that if I didn't understand it I should drop the course and take the foundation math course which was noncredit. Well it was past the drop/add deadline and I couldn't afford to take this course again. Failing was not an option! So before the first exam I went to the library and booked one of those little study rooms. I tried to do the problems we had worked on in class, but always found myself getting the same wrong answer. The frustration would build and I can still see myself slamming books down and flipping pages wildly, stamping out of the room with a " I can't do this" sense of fatalism. Then I'd make myself go back and try again and a strange thing happened on the way to giving up, a light went on. I figured out what I was doing wrong. What elation, what a feeling of accomplishment what a change in my view of math! Math was I realized something I could do if I was willing to work long and hard. Not knowing the answer right away no longer meant "I can't do this" it meant that I had to keep working.
Giving up was the way I had always faced math. Math teachers in general didn't give me a lot of time, and it felt as if they seemed to agree that I was beyond help and so very little was ever given ( although to be honest this was more of subconscious feeling) . Except in grade 10 when I had a math teacher who expected me to work hard, who checked to see if my homework was done, who kept me after to class to grill me if it wasn't and who called on me to answer questions as regularly as he did anyone else. This feeling of having to live up to his standard was in equal parts frustrating and motivational. He was the best! The fact that he took his time, that he actually looked at the class when asking if anyone needed help instead of having his back to the room while erasing the board were all factors in making him stand out as a good math teacher. He loved the subject and he liked us and everyone knew it.
This could not be said about any of my other high school math teachers. Assignments usually consisted of pages of repeated practice problems. Similar to regular math homework but printed in hand on a mimeographed sheet. Assessment consisted of marks given for end of unit tests, some small weight given for homework completion, assignments were marked and of course the ubiquitous mid-term and final exams. Summative evaluation was the norm for all the years I was in school and was entrenched as the way things were done. This pattern continued into university through the 4 math courses I took while there. The courses at university consisted of the two first year entrance level courses, Primary/ Elementary Math 1 and 2 and the Math Methods course. I may have had that epiphany but I still didn't actually like math.
So it continued, I taught math in many grades as a substitute and replacement teacher and felt I was doing a good job. I did what all the other teacher seemed to be doing, what my teachers did. I taught, they listened, they set to work on problems. This was how it was until they brought in a new math program and it made its way to grade 5 several years ago. It was a totally new approach to teaching and assessing students. Assessment became multi-layered and the problem solving approach to mathematics was de rigeur. What horror, oh the complaining and gnashing of teeth in the staffroom. I struggled to make sense of it while clinging to the ways I had always taught. But the struggle became to big and I gave myself over to change. I have put in a tremendous amount of time into changing my approach to the teaching of math and I can see the difference this program has made to so many students. Having said that I must add that in my opinion no one approach can meet all the needs. Differentiation sounds like it may hold the key and is something that I have included in my professional growth plan. These changes in my attitude have developed over a good many years. I have gone from hopeless to hopeful. I feel that mathematics can be an exciting and rewarding experience for all students. While I struggle to make it so, I know that time and effort pays off and perserverence is the name of the game!
Monday, September 14, 2009
Critical Reflections Dialogue Blog 1
Thursday, September 10
Class response
As an introduction to the current issues of significance in mathematics education, and for me an introduction to the world of graduate courses, this first class was an eye-opener. I enjoyed our surroundings and our conversation and am looking forward to future classes!
Frozen Future: I was struck by this phrase and have discussed with colleagues and family the significance this has for all education really! We are involved in educating students for a future which existed when we were younger but no longer does. In our experience as students ourselves and as educators it is ingrained in us that teachers are in control of information that students need, they deconstruct the needed information and rebuild for students and if those students didn't get it they weren't paying attention. This pedagogical background then encounters students whose needs or skill set in the digital age are a mismatch with what we have always believed they needed to know. Alvin Toffler, an American writer and futurist, in his book 'Rethinking the Future' says "The illiterate of the 21st century will not be those who cannot read or write but those who can not learn, unlearn and relearn." Certainly this issue gave me long pause for thought, and will help inform what I see as critical issues in mathematical education.
As an introduction to the current issues of significance in mathematics education, and for me an introduction to the world of graduate courses, this first class was an eye-opener. I enjoyed our surroundings and our conversation and am looking forward to future classes!
Frozen Future: I was struck by this phrase and have discussed with colleagues and family the significance this has for all education really! We are involved in educating students for a future which existed when we were younger but no longer does. In our experience as students ourselves and as educators it is ingrained in us that teachers are in control of information that students need, they deconstruct the needed information and rebuild for students and if those students didn't get it they weren't paying attention. This pedagogical background then encounters students whose needs or skill set in the digital age are a mismatch with what we have always believed they needed to know. Alvin Toffler, an American writer and futurist, in his book 'Rethinking the Future' says "The illiterate of the 21st century will not be those who cannot read or write but those who can not learn, unlearn and relearn." Certainly this issue gave me long pause for thought, and will help inform what I see as critical issues in mathematical education.
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