Mathematical empowerment. What a loaded phrase from chapter 8. What does it mean to be “mathematically empowered”? Boaler suggests that the students at Phoenix Park were mathematically empowered because they were flexible in their approach to mathematics, because they were prepared to take what they had learned and adapt it to new situations. The flexibility in their thinking and approach to mathematics was reliant on two key features, one being that students believed the math they were learning was adaptable, that it had many uses and could be applied in numerous situations and secondly it relied on students ability to adapt and change the math methods, to think mathematically. These students do not see any separation between the mathematics they are learning at school and the mathematics they are using in real life, they feel that they have knowledge and skills that are of use to them, as opposed to being in a possession of fragmented and disconnected ideas about mathematics, much like the Amber Hill students.
Who we are as teachers will determine who our students become as learners and users of mathematics. If we provide them with challenges and aim to have them construct meaning for themselves then our students are likely to develop this sense of “empowerment” that Boaler describes in the Phoenix Park students. On the contrary, if we provide our students with set procedures and methods, rules that they must follow without having a true understanding of the mathematics behind “the rules” then we are likely to produce students of a similar calibre of those at Amber Hill. Students who are led to believe that they will do well in mathematics if they “follow the rules” but who find that in the real world they are not prepared to think about and make sense of problems of a mathematical nature, students who have a false sense of security that has been provided by cuing systems, rules and practice.
I know that I would like to prepare my students to face “the real world” of mathematics; I would like them to approach problems with the certainty that they are capable of arriving at a solution through their own thinking (without having to rely on long forgotten rules). I would like my students to believe in themselves as learners and users of mathematics but I would also like them to believe in the mathematics. In constructing their own knowledge, taking time to investigate and make sense, taking time to understand, time to see why “the rules” work without just practicing rules. I believe students can become more skilled in their use of school mathematics, whether it is for school use of real world use. How can I achieve this? How can I ensure that my students are not just becoming passive users of mathematics, that they are not simply “playing the game” or “following the rules” much like the students at Amber Hill? I think that even at a young age students can assume some of the responsibility for the construction of knowledge, knowledge that is built through hands on investigations rather than worksheets or text book questions that follow a chalk and talk session directed by the teacher. Can we really expect young students to grasp some of mathematical concepts at a symbolic level when they have not had enough time to experiment with it at a concrete level? We see from Boaler’s work that it didn’t work at Amber Hill, so it is likely to fail our own students as well.
Like the teachers at Amber Hill and Phoenix Park I too have my student’s best interest at heart when I plan my mathematics lessons. The challenge now is to combine best interest with best practice, to use instructional methods that allow students to construct knowledge and apply knowledge, in the classroom and in the real world.
Saturday, November 21, 2009
Sunday, November 15, 2009
Chapter 7 - Exploring the Differences......
What stuck with me the most from this chapter was the false sense of success the Amber Hill students must have experienced in mathematics, particularly those in the higher sets. Students worked through the classroom assignments and textbook work with little difficulty, if they followed the steps and memorized the rules then surely they could do well in maths. However, when the time came for the GCSE’s and the questions were posed in an unfamiliar way, when cuing systems were removed and the math had to be applied as opposed to memorized the Amber Hill students found they were ill-prepared for the test. They realized that when their memory failed them they had little else to turn to; they had no real understanding of the math.
On the flip side of this coin we have the Phoenix Park students, while they may not have been prepared for the test entirely in that there were some topics that they had not encountered or been taught through their project work, these students still fared better in the GCSE’s because they could think about the math. These students could make the connections between mathematics and its real world applications. They did not rely on the cuing systems which Amber Hill students struggled without. They were more willing to try questions when the answer or method wasn’t instantly apparent.
Towards which end of this spectrum are our own students lying? Are they prepared for new and challenging tasks? Are they willing to take risks in mathematics to increase their understanding or are they reliant on us as teachers to provide them with the answers to “difficult questions”?
How often do we hear our colleagues talking about classes that just don’t “get it”, classes that don’t “know their basic facts”? Why is it that year after year we hear these same questions, even though we know that we have taught these skills and ideas? Amber Hill’s teachers were especially concerned with their students’ success, so much so that they provided all sorts of “helpful hints” to get them through the math, all sorts of rules that they “just had to remember” and we can see in the end how these students fared. Let’s set things up differently for our students, let’s encourage them to think, to make sense on their own and to truly understand the mathematics we are “teaching”.
On the flip side of this coin we have the Phoenix Park students, while they may not have been prepared for the test entirely in that there were some topics that they had not encountered or been taught through their project work, these students still fared better in the GCSE’s because they could think about the math. These students could make the connections between mathematics and its real world applications. They did not rely on the cuing systems which Amber Hill students struggled without. They were more willing to try questions when the answer or method wasn’t instantly apparent.
Towards which end of this spectrum are our own students lying? Are they prepared for new and challenging tasks? Are they willing to take risks in mathematics to increase their understanding or are they reliant on us as teachers to provide them with the answers to “difficult questions”?
How often do we hear our colleagues talking about classes that just don’t “get it”, classes that don’t “know their basic facts”? Why is it that year after year we hear these same questions, even though we know that we have taught these skills and ideas? Amber Hill’s teachers were especially concerned with their students’ success, so much so that they provided all sorts of “helpful hints” to get them through the math, all sorts of rules that they “just had to remember” and we can see in the end how these students fared. Let’s set things up differently for our students, let’s encourage them to think, to make sense on their own and to truly understand the mathematics we are “teaching”.
Saturday, November 7, 2009
Who's really in charge here......
Discussions in class lead me to question who is really in charge of the learning in our classrooms? If we are supposed to meet our students at their level and take them through the curriculum over the course of the school year, why then are school districts imposing rigid schedules and outlines for us to follow in our classrooms? What if our students are not ready, what if they need more time? How can we truly do justice to a math program and our students math education if we are so caught up with "staying on track" of a schedule imposed from above, the schedule should come from the grassroots level, from the students themselves. It only stands to reason that if we move on to newer and more complicated areas of mathematics before our students are ready, before they have a good understanding of "the basics" that they will continue to struggle, that they will never "catch up", that they will never enjoy mathematics or come to see it's usefulness.
I have to say I never experienced this strict regime when teaching in Alberta, nobody imposed a schedule on me, and I was actually encouraged to spend additional time on areas where students were struggling, knowing that in the end it would allow my students to grow in mathematics as a good foundation had been put in place. Nobody ever told me what HAD to be assessed at each reporting period, I also exercised choice over the resources and assessment methods used within my classroom, if the other grade level teacher chose to give pencil and paper tests and I chose to assess in a different manner that was ok, we did what worked for us, and for our students. Here we have teachers who not only must assess in the same manner, but on the same day.......whatever happened to differentiation? Since when does one size fits all work for students? for teachers?
If we are to provide our students with a truly rich mathematical experience then we must tailor that experience to meet their needs and to meet our needs as teachers. We need to make the choices that are best for our students, choices about when and how to assess, how quickly to move through the curriculum, choices about the types of activities and resources we use, choices that allow our students to make sense of mathematics and build their own understandings of concepts; as opposed to rushing through a series of text book lessons, expecting students to make sense of abstract symbols and concepts without the hands on, concrete experiences that although they may take more time and effort to complete can provide our students with a much deeper understanding than a teacher provided demonstration and pages of pencil and paper practice.
It's time to give the control back to those who need it, the students. Let them dictate the speed of our lessons, the activities we use and the assessment tasks. Let's give them the opportunities they deserve, the opportunities that many of us were not afforded during our own math education.
Melanie
I have to say I never experienced this strict regime when teaching in Alberta, nobody imposed a schedule on me, and I was actually encouraged to spend additional time on areas where students were struggling, knowing that in the end it would allow my students to grow in mathematics as a good foundation had been put in place. Nobody ever told me what HAD to be assessed at each reporting period, I also exercised choice over the resources and assessment methods used within my classroom, if the other grade level teacher chose to give pencil and paper tests and I chose to assess in a different manner that was ok, we did what worked for us, and for our students. Here we have teachers who not only must assess in the same manner, but on the same day.......whatever happened to differentiation? Since when does one size fits all work for students? for teachers?
If we are to provide our students with a truly rich mathematical experience then we must tailor that experience to meet their needs and to meet our needs as teachers. We need to make the choices that are best for our students, choices about when and how to assess, how quickly to move through the curriculum, choices about the types of activities and resources we use, choices that allow our students to make sense of mathematics and build their own understandings of concepts; as opposed to rushing through a series of text book lessons, expecting students to make sense of abstract symbols and concepts without the hands on, concrete experiences that although they may take more time and effort to complete can provide our students with a much deeper understanding than a teacher provided demonstration and pages of pencil and paper practice.
It's time to give the control back to those who need it, the students. Let them dictate the speed of our lessons, the activities we use and the assessment tasks. Let's give them the opportunities they deserve, the opportunities that many of us were not afforded during our own math education.
Melanie
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