The results presented in this chapter brought to light the reality that the traditional teaching methods employed by the mathematics teachers at Amber Hill proved to be of disservice to the students when the time came to show what they know.....which in fact was very little. Students demonstrated that they actually understood very little of what they had covered in math lessons and heavily relied on memory and on obscure cueing systems to help them to figure out the answers to both procedural and conceptual problems. When memory and cueing systems failed them students had no idea how to go about solving the problems, no idea how to make sense of it on their own, and in many cases couldn’t even make sense of what the question was asking.
On the flip side of this coin, students at Phoenix Park had been exposed to a dramatically different teaching method, one which relied heavily on students own understanding of topics and their own creation of knowledge. It may appear that some students were disadvantaged in that they hadn’t been exposed to some mathematical formulas or “rules” and many did not have calculators to complete the exam, however, they were prepared to face the exam with a much more powerful and useful tool than a calculator or list of disconnected and misunderstood rules. These students were prepared to approach problems without fear, they were willing to take risks, and they were willing to THINK! Phoenix Park students had experienced a type of mathematics in which the first answer wasn’t always the right answer, in which the right answer wasn’t necessarily one that could be found in 2 or 3 minutes. Though they may not have had “practiced” numerous examples in preparation for the GCSE’s, the problems they had worked on all year had provided them with practice in thinking, in understanding, in being creative and in finding a way to solve problems that makes sense to them. They had become more skilled at thinking about the math than in thinking about the answer.
How wonderful it would be if we could prepare our students to face exams such as Publics, common finals and CRT’s with the same approach as the teachers at Phoenix Park, if we could in effect stop teaching to the test and start teaching to the students. Meeting the students where they are instead of dragging them to where we want them to be, using their existing knowledge as the basis for future knowledge and having them really understand the math as opposed to sort of remembering it (because we know they don’t remember it for very long.) What would it take to get to this point? Time, resources and a willingness to step back and evaluate our own instructional methods, our own ideas about what good math teaching looks like, a willingness on our part to accept change and make change.
Melanie
Saturday, October 31, 2009
Wednesday, October 28, 2009
Intent vs. Impact
In reading the chapters on Amber Hill and Phoenix Park one idea became remarkably clear, the intent of our teaching does not always have the desired impact. The teachers at Amber Hill cared about their students’ success, and chose their teaching methods as their way to bring about maximum success in mathematics. They felt that the structure of the lessons and the division of students in sets allowed for maximum opportunities for students to be successful. Teachers at Phoenix Park also were concerned with their students’ achievement and used a remarkably different set of teaching strategies to ensure that students were successful. The connection between intent and impact was there at Phoenix Park, but Amber Hill’s intent and impact were disjointed. Students at Amber Hill were not successful, they had difficulty with mathematics not only in test situations but in real world workings as well, they saw math as rule governed and isolated, and assumed that if you couldn’t remember the rules or figure out which rule to use, then you had little chance of succeeding in mathematics. Phoenix Park students on the other hand were far more successful in both test situations and real world application of math skills, they saw the connections and even if they did not immediately have a strategy to use they were able to work through problems and projects building on existing knowledge.
How often do we teach something, assess it, and although a large number of students may still be struggling with a concept we move on, hoping perhaps that they will eventually get it. The intent of our teaching has to change from having our students “get it” to having our students understand it and use it. Skills in isolation are of little use to our students, they must be given frequent and varied opportunities to use these skills, to think about the connectedness of mathematics and forget the rules and think about it on their own.
What was it a wise man once said about good intentions......
How often do we teach something, assess it, and although a large number of students may still be struggling with a concept we move on, hoping perhaps that they will eventually get it. The intent of our teaching has to change from having our students “get it” to having our students understand it and use it. Skills in isolation are of little use to our students, they must be given frequent and varied opportunities to use these skills, to think about the connectedness of mathematics and forget the rules and think about it on their own.
What was it a wise man once said about good intentions......
Tuesday, October 27, 2009
Phoenix Park - Breaking the "Rules"
Openness, choice, freedom, independence; characteristics that many would agree are conducive to a successful classroom, a classroom which promotes learning and encourages students to meet their full potential. Why then were these characteristics so obviously noticeable at Phoenix Park and so obviously missing at Amber Hill? One thing is certain, both schools were concerned with the success of their students in mathematics, yet the schools used remarkably different approaches in an attempt to bring about this success.
Phoenix Park, remarkably different from Amber Hill, and perhaps if we were to investigate we would find that the approach to learning was in fact very different from much of what is presently happening in our own schools.
Michelle raised some interesting questions in her presentation, of particular interest is the question, is this possible here? The question is raised in reference to the freedom and openness experienced at Phoenix Park which is unlike most school settings. The obvious answer is yes, of course it is possible, Boaler has shown that it was possible to achieve this learning style in a school in a working class neighbourhood, with students who had previously been exposed to more traditional methods of instruction, schools much like our own! The evidence in Boaler’s study clearly indicates that such an approach CAN work and CAN be successful for ALL students, how then can be incorporate these ideas into our own classrooms? Surely we cannot be expected to make an overnight switch from a method that we have become accustomed to over many years, but even small changes will help our students start to develop a deeper level of mathematical understanding.
Our current classroom situation is much like Amber Hill’s, and we are heavily influenced by “higher powers” and CRT achievement. This HAS to change; the driving force of classroom instruction needs to be the students, and not the numbers that show how a group of students performed on a series of test questions. As we have discussed in class many numbers, the interpretation of these results is driving our instruction, but the results are not being interpreted in a way that makes sense. Schools are comparing results from one year to the next as opposed to comparing a group of students in Grade 3 and then following up in Grade 6 and 9 and comparing them to their own achievement and growth. However, if the idea of following up involves drilling our students with more “practice” exercises and “practice” tests in areas that were deemed to be areas of weakness in the test, then I see no need to even look at the results if we are not going to look at our teaching methods as well. We need to take a serious look at not just WHAT we are teaching, but HOW we are teaching it.
Michelle also raised the question of “What is structure?” I think we can all agree that structure is necessary for a successful classroom environment, but what does structure consist of? Is it merely the physical arrangement of the classroom? Does it include the teachers role within the classroom, the students and their actions? If we are to include all of these elements then we can clearly see that the “structure” at Phoenix Park and Amber Hill were again, remarkably different. Amber Hill relied on the teacher to maintain structure, and we know from Boaler’s findings that this so called structure did not necessarily indicate that students were spending any more time on task than at Phoenix Park, though students at Amber Hill were very well trained in playing the game, in looking like they were working.
To an outsider (or even to a teacher from Amber Hill) Phoenix Park’s math classrooms may have looked like they were lacking structure, but that is not necessarily the case. The classrooms were arranged in such a way that students were able to work on their own or in groups to work on the assigned projects, and although the teacher did not direct the entire lesson, they did provide guidance to students as they needed it, although they did not “make” students work, but really, are we actually able to do this? Can we force students to think? At Phoenix Park students were very much aware of their responsibilities and aware of the consequences that would follow if they did not complete the assigned tasks, some still wasted class time, yet this is a likely scenario in any class, despite our best efforts to “structure” the learning experience. Phoenix Park did a much better job of providing for students at a range of abilities, and all within the one classroom/lesson; something Amber Hill couldn’t accomplish even when they divided their students in sets according to ability, students still complained that they were either over loaded with work or not sufficiently challenged.
I would have loved to have been a student at Phoenix Park, to have been afforded the opportunity to make sense of mathematical ideas on my own, to make connections and develop an understanding which would have stuck with me long past the unit tests (who am I kidding, I never even understood most things then, but I could remember enough of the rules for the tests.) My past experience with math encourages me as a teacher rather than discourages me. I am guilty of having an “unstructured” classroom at times, I don’t mind noise, and mess is my friend, I know I have a long way to come as a math teacher but I feel that I have made great strides in allowing my students to experience greater freedom in mathematics, in allowing them to find the answers the best way they know how as opposed to memorizing my way. Math is not just something that comes from a textbook and there are far more ways to demonstrate understanding than pencil and paper tests, we need to embrace these ideas and put them into practice to provide our students with rich mathematical experiences.
Phoenix Park, remarkably different from Amber Hill, and perhaps if we were to investigate we would find that the approach to learning was in fact very different from much of what is presently happening in our own schools.
Michelle raised some interesting questions in her presentation, of particular interest is the question, is this possible here? The question is raised in reference to the freedom and openness experienced at Phoenix Park which is unlike most school settings. The obvious answer is yes, of course it is possible, Boaler has shown that it was possible to achieve this learning style in a school in a working class neighbourhood, with students who had previously been exposed to more traditional methods of instruction, schools much like our own! The evidence in Boaler’s study clearly indicates that such an approach CAN work and CAN be successful for ALL students, how then can be incorporate these ideas into our own classrooms? Surely we cannot be expected to make an overnight switch from a method that we have become accustomed to over many years, but even small changes will help our students start to develop a deeper level of mathematical understanding.
Our current classroom situation is much like Amber Hill’s, and we are heavily influenced by “higher powers” and CRT achievement. This HAS to change; the driving force of classroom instruction needs to be the students, and not the numbers that show how a group of students performed on a series of test questions. As we have discussed in class many numbers, the interpretation of these results is driving our instruction, but the results are not being interpreted in a way that makes sense. Schools are comparing results from one year to the next as opposed to comparing a group of students in Grade 3 and then following up in Grade 6 and 9 and comparing them to their own achievement and growth. However, if the idea of following up involves drilling our students with more “practice” exercises and “practice” tests in areas that were deemed to be areas of weakness in the test, then I see no need to even look at the results if we are not going to look at our teaching methods as well. We need to take a serious look at not just WHAT we are teaching, but HOW we are teaching it.
Michelle also raised the question of “What is structure?” I think we can all agree that structure is necessary for a successful classroom environment, but what does structure consist of? Is it merely the physical arrangement of the classroom? Does it include the teachers role within the classroom, the students and their actions? If we are to include all of these elements then we can clearly see that the “structure” at Phoenix Park and Amber Hill were again, remarkably different. Amber Hill relied on the teacher to maintain structure, and we know from Boaler’s findings that this so called structure did not necessarily indicate that students were spending any more time on task than at Phoenix Park, though students at Amber Hill were very well trained in playing the game, in looking like they were working.
To an outsider (or even to a teacher from Amber Hill) Phoenix Park’s math classrooms may have looked like they were lacking structure, but that is not necessarily the case. The classrooms were arranged in such a way that students were able to work on their own or in groups to work on the assigned projects, and although the teacher did not direct the entire lesson, they did provide guidance to students as they needed it, although they did not “make” students work, but really, are we actually able to do this? Can we force students to think? At Phoenix Park students were very much aware of their responsibilities and aware of the consequences that would follow if they did not complete the assigned tasks, some still wasted class time, yet this is a likely scenario in any class, despite our best efforts to “structure” the learning experience. Phoenix Park did a much better job of providing for students at a range of abilities, and all within the one classroom/lesson; something Amber Hill couldn’t accomplish even when they divided their students in sets according to ability, students still complained that they were either over loaded with work or not sufficiently challenged.
I would have loved to have been a student at Phoenix Park, to have been afforded the opportunity to make sense of mathematical ideas on my own, to make connections and develop an understanding which would have stuck with me long past the unit tests (who am I kidding, I never even understood most things then, but I could remember enough of the rules for the tests.) My past experience with math encourages me as a teacher rather than discourages me. I am guilty of having an “unstructured” classroom at times, I don’t mind noise, and mess is my friend, I know I have a long way to come as a math teacher but I feel that I have made great strides in allowing my students to experience greater freedom in mathematics, in allowing them to find the answers the best way they know how as opposed to memorizing my way. Math is not just something that comes from a textbook and there are far more ways to demonstrate understanding than pencil and paper tests, we need to embrace these ideas and put them into practice to provide our students with rich mathematical experiences.
Wednesday, October 14, 2009
Amber Hill - Keeping with Tradition
Kudos to Sharon for her fantastic presentation last week!
You pointed out many important ideas from the text and brought about an engaging conversation. Some of the main ideas that came through for me (and that I am still trying to make sense of or understand a little more) are:
• The difference that exists between what we “believe” good math teaching is, and how we present this so-called “good teaching” to teaching. It is safe to say that educators, including those at Amber Hill, would consider student understanding to be essential to “good teaching”. However, for the most part students in math classes at Amber Hill were not understanding, they were simply “getting on with it” or “going through the motions”, students were completing numerous practice questions in class and recreating these same types of questions on tests, yet when students had to take these ideas and apply them in context or in a way other than which they were used to they couldn’t do it, they didn’t understand...... which certainly leads to the question, did they ever understand it? The answer is an alarming “no”; the students themselves are reporting that they are simply “following the rules”. If the students themselves believe that the most important part of math is remembering, how then can we expect them to use these skills in future situations when the formulas and rules are long forgotten and they don’t have the understanding necessary to derive these “rules” on their own?
• Teachers at Amber Hill recognize that students are not developing a true understanding of mathematical concepts and are unable to transfer their knowledge (“Tim: Students are generally good unless a question is slightly different to what they are used to, or if they are asked to do something after a time lapse, if a question is written in words or if they are expected to answer in words.” p. 34). If teachers are recognizing that students do not understand, what are they blaming this lack of understanding on? Teachers are quick to point out factors related to the students background, time on task, etc, but do not readily examine their own teaching practice as a possible contributor. Why is this? Are we afraid to admit that we perhaps don’t have all of the answers? We need to see a pedagogical shift emerge that supports the idea of “Have they learned it? If not, how can I help them to learn it?” as opposed to the view of “I taught it so they better have learned it.”
• Student success appears to be one of the forces driving instruction at Amber Hill, yet the success is not evident. Teachers “care deeply about their students” yet this level of care is not such that it translates into the methods of instruction that are selected for students. Teachers rely heavily on a traditional method of instruction, they are confident that the level of structure provided by the textbooks will serve their students better. Students however, are more satisfied with the learning taking place when working on the open ended course work, something which they only experience for 2-3 weeks each year beginning at the end of year 9. There is an obvious disconnect here in what students believe to be important to mathematical understanding and what teachers believe, more opportunities for open ended work and student dialogue would serve to bridge this gap and produce greater understanding.
I would suspect that much of what is happening at Amber Hill is happening right here at home. Teachers here, there, everywhere are concerned with the success of their students, but the instructional methods being used in our classrooms are not necessarily delivering that level of success that we would like to see.
A host of reasons can be suggested as to why these instructional methods are still heavily relied upon many years after studies have proven them to be not nearly as effective as the hands-on student centered approaches Boaler describes at Phoenix Park School. Curriculums are loaded with objectives and time is something there is never enough of, teachers are pressured to complete the curriculum in a “suggested” time-frame.
Money.....something else often in short supply, with large class sizes and a lack of manipulatives in many classrooms, completing lessons that stray from the traditional approach described at Amber Hill can certainly pose a challenge.
The traditional approach is one that has been tried and tested in our classrooms for many years, and while it is perhaps not providing our students with maximum mathematical understanding, it is not outright failing them (when it comes to standardized testing anyway.....) so as teachers we may be inclined to stick with what we know (as this is how many of us ourselves were taught.)
CRT’s much like standardized tests taken by students at Amber Hill have a drastic effect on instruction at the grade levels in which they are administered. There is a tremendous amount of pressure on teachers to have their students meet certain expectations, and the teachers are in turn putting this pressure on students to perform. What is the purpose of these tests? What do they really show us about student understanding? Can a multiple choice test that has in many ways been rehearsed really show us what our students truly know and understand? Is this how we define success in mathematics, the ability to do well in a standardized test situation?
Whatever the reason for the inadequacies in Amber Hill’s mathematics program and in our own mathematics programs in Newfoundland and Labrador, it is obvious that student success must be first and foremost our goal. Defining what it means to be successful at mathematics will perhaps lead us to change within our instructional strategies.
You pointed out many important ideas from the text and brought about an engaging conversation. Some of the main ideas that came through for me (and that I am still trying to make sense of or understand a little more) are:
• The difference that exists between what we “believe” good math teaching is, and how we present this so-called “good teaching” to teaching. It is safe to say that educators, including those at Amber Hill, would consider student understanding to be essential to “good teaching”. However, for the most part students in math classes at Amber Hill were not understanding, they were simply “getting on with it” or “going through the motions”, students were completing numerous practice questions in class and recreating these same types of questions on tests, yet when students had to take these ideas and apply them in context or in a way other than which they were used to they couldn’t do it, they didn’t understand...... which certainly leads to the question, did they ever understand it? The answer is an alarming “no”; the students themselves are reporting that they are simply “following the rules”. If the students themselves believe that the most important part of math is remembering, how then can we expect them to use these skills in future situations when the formulas and rules are long forgotten and they don’t have the understanding necessary to derive these “rules” on their own?
• Teachers at Amber Hill recognize that students are not developing a true understanding of mathematical concepts and are unable to transfer their knowledge (“Tim: Students are generally good unless a question is slightly different to what they are used to, or if they are asked to do something after a time lapse, if a question is written in words or if they are expected to answer in words.” p. 34). If teachers are recognizing that students do not understand, what are they blaming this lack of understanding on? Teachers are quick to point out factors related to the students background, time on task, etc, but do not readily examine their own teaching practice as a possible contributor. Why is this? Are we afraid to admit that we perhaps don’t have all of the answers? We need to see a pedagogical shift emerge that supports the idea of “Have they learned it? If not, how can I help them to learn it?” as opposed to the view of “I taught it so they better have learned it.”
• Student success appears to be one of the forces driving instruction at Amber Hill, yet the success is not evident. Teachers “care deeply about their students” yet this level of care is not such that it translates into the methods of instruction that are selected for students. Teachers rely heavily on a traditional method of instruction, they are confident that the level of structure provided by the textbooks will serve their students better. Students however, are more satisfied with the learning taking place when working on the open ended course work, something which they only experience for 2-3 weeks each year beginning at the end of year 9. There is an obvious disconnect here in what students believe to be important to mathematical understanding and what teachers believe, more opportunities for open ended work and student dialogue would serve to bridge this gap and produce greater understanding.
I would suspect that much of what is happening at Amber Hill is happening right here at home. Teachers here, there, everywhere are concerned with the success of their students, but the instructional methods being used in our classrooms are not necessarily delivering that level of success that we would like to see.
A host of reasons can be suggested as to why these instructional methods are still heavily relied upon many years after studies have proven them to be not nearly as effective as the hands-on student centered approaches Boaler describes at Phoenix Park School. Curriculums are loaded with objectives and time is something there is never enough of, teachers are pressured to complete the curriculum in a “suggested” time-frame.
Money.....something else often in short supply, with large class sizes and a lack of manipulatives in many classrooms, completing lessons that stray from the traditional approach described at Amber Hill can certainly pose a challenge.
The traditional approach is one that has been tried and tested in our classrooms for many years, and while it is perhaps not providing our students with maximum mathematical understanding, it is not outright failing them (when it comes to standardized testing anyway.....) so as teachers we may be inclined to stick with what we know (as this is how many of us ourselves were taught.)
CRT’s much like standardized tests taken by students at Amber Hill have a drastic effect on instruction at the grade levels in which they are administered. There is a tremendous amount of pressure on teachers to have their students meet certain expectations, and the teachers are in turn putting this pressure on students to perform. What is the purpose of these tests? What do they really show us about student understanding? Can a multiple choice test that has in many ways been rehearsed really show us what our students truly know and understand? Is this how we define success in mathematics, the ability to do well in a standardized test situation?
Whatever the reason for the inadequacies in Amber Hill’s mathematics program and in our own mathematics programs in Newfoundland and Labrador, it is obvious that student success must be first and foremost our goal. Defining what it means to be successful at mathematics will perhaps lead us to change within our instructional strategies.
Tuesday, October 6, 2009
Reflecting....3,2,1
After reading Shoenfeld’s Introduction and Chapters 1-3 of Boaler’s book I was left with an alarming feeling of déjà vu. I felt as though I had been a student at Amber Hill, sitting alongside the students working through those same textbook exercises after listening to the teacher explain the procedure to be followed. Having been taught in a traditional way, much like the teaching methods at Amber Hill, I wasn’t surprised to read in the introduction that students at Phoenix Park outperformed those from Amber Hill in standardized tests, tests in which students were to take what they knew about mathematics and apply it in new and unusual circumstances. Amber Hill students were programmed to take a formula/procedure, memorize it, and plug in new numbers to find an answer.....without ever really understanding where the formula or procedure came from or why they were using it.
Students at Phoenix Park on the other hand spent a great deal of time using what they knew in mathematics to help them work towards an understanding of new concepts, they worked from the known to the unknown using their own knowledge and experiences as a guide in their discovery. Students at Phoenix Park were the driving force of the instruction in their classrooms, the goal of math lessons was understanding. Students were encouraged to think about and understand the mathematics on their own; the teachers provided guidance to the students as they worked on open ended tasks, but the students did not rely on the teachers to help them find the right answer as did students at Amber Hill.
I’m interested in reading more about the math instruction in each of these schools, I am already wishing that I had been a “Phoenix Park” student rather than an “Amber Hill” student.
Melanie
Students at Phoenix Park on the other hand spent a great deal of time using what they knew in mathematics to help them work towards an understanding of new concepts, they worked from the known to the unknown using their own knowledge and experiences as a guide in their discovery. Students at Phoenix Park were the driving force of the instruction in their classrooms, the goal of math lessons was understanding. Students were encouraged to think about and understand the mathematics on their own; the teachers provided guidance to the students as they worked on open ended tasks, but the students did not rely on the teachers to help them find the right answer as did students at Amber Hill.
I’m interested in reading more about the math instruction in each of these schools, I am already wishing that I had been a “Phoenix Park” student rather than an “Amber Hill” student.
Melanie
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