When I finished reading Shoenfeld’s article, my very first thought was of good teaching, what is good teaching, and who defines it? In his article Schoenfeld cites researchers Romberg and Carpenter to point to the fact that much of the instruction in schools at that time (and currently if I had to guess) was based on the “absorbtion theory of learning”. This theory is based on the premise that good teaching is defined by having many ways to say the same thing so that students will eventually “get it”.
Absorbtion theory, chalk and talk, kill and drill, call it what you like, students need to make sense of mathematical concepts on their own if they are to develop a lasting understanding of an idea.
Case in point, I had a tutoring student this week, who, although I have stopped tutoring her now that she is in Gr 7, I agreed to see her once more to help her get ready for her first test of the year. The topic being tested was divisibility rules and patterns, a skill that, if understood, can certainly be useful in completing more challenging mathematical tasks. We took some time working with the other rules, which she had no trouble with, she could write numbers that were divisible by a given number, check to see if a number was divisible by a given number and list all of the divisibility rules that applied to a number........she could apply all of the rules....except for the rule for dividing by 8. When it came to the divisibility rule of 8, “A number is divisible by 8 if its last three digits is divisible by 8” this student was having real difficulty......and with good reason I thought....I mean really, who knows their multiplication tables up to 3 digit products, yet this is the “rule” that she was supposed to use (and incidentally is the rule in the text book). I asked her what she knew about the number 8, what was it divisible by.....she quickly came up with 4 and 2, I asked her if she thought this could be of any help to her in determining if a number was divisible by 8. She didn’t see a connection right away, but when she did some more work with the divisibility rule for 6 ( a number is divisible by 6 if it is divisible by both 2 and 3), she was immediately able to make the connection that a number that is divisible by 8 is divisible by 4 and then by 2, for example if 328 is divided by 4 the answer is 82, 82 is divisible by 2, therefore 328 is divisible by 8. 292 is divisible by 4, but the resulting answer is 73, which is not divisible by2, therefore 292 is not divisible by 8. It took some time, and a little trial and error, but eventually she was able to use one rule to help her come up with another.
She used her understanding of one rule to develop her own understanding of another, she made sense of it on her own, and has a more reliable way to find a solution as opposed to applying a rule that she doesn’t understand and doesn’t have the skills to use. Had she been forced to rely on this rule she had been “taught” it is likely that she would not be able to apply it and would not demonstrate an understanding of something that she does in fact understand quite well......when she’s given the time to think about it.
Another question that came to mind while I was reading this article is whether or not we are meeting the long term needs of our students with instruction that is intended to produce short term results? In the school in which Schoenfeld conducted his 1983-84 study, “Learning was operationally defined as performance on achievement tests” (Schoenfeld p. 4), “the primary goal of instruction was to have students do well on the exam” (Schoenfeld p.9.)
In placing so much emphasis on these tests we are influencing the very activities which take place in our classrooms. The phrase “teaching to the test” is one that is echoed by teachers everywhere, often with contempt, but is one that many adhere to even though they may be in disagreement. Our school districts are adopting new curriculums and new instructional materials that emphasize learning for understanding, yet we are still using materials for testing that emphasize memorization. I think that our assessment methods have to come in line with our instructional methods before we are able to see any real change in producing students who are able to apply their mathematical understanding in different and new situations.
On page 16 Schoenfeld states that what he observed in his study was “typical, if not better than average, instruction—with typical results” , as long as standardized test scores remains one of the driving forces of math instruction in our schools I feel it is likely that we will continue to see this “typical situation.
Melanie
Monday, September 28, 2009
Saturday, September 19, 2009
Mathematics Autobiography.....
In reflecting on my early math education I can see that that it was very “traditional” in nature, the perception that “practice makes perfect” is one that I am sure was evident in many (if not most) Newfoundland classrooms throughout the 1980’s and early 90’s.
From K-6 Math instruction in my classroom generally involved pencil and paper, the manipulatives were present in the classroom, but rarely made their way into lessons and when they did we were so in awe of these “blocks” that we were not permitted to “play” with at any other time, that little was ever really accomplished in terms of mathematics within these lessons. The teacher would use the chalkboard to do a few examples of whatever skill or strategy we were working on that day and then we would set to work on pages of practice. I remember getting new workbooks in Grade 2 as part of the MathQuest series, I also remember not doing any of the pages that involved cutting and pasting as these pages were "too messy and wasted too much time." In trying to find a way to describe the role of the teacher in my early math classes I came across the following Ten Roles for Math Teachers. I don't think any of these describe the math classrooms I experienced early in my schooling. I think it could best be described as teacher as demonstrator. The teacher demonstrated a skill, we were set about on our own to practice it, if we didn't understand it the teacher would demonstate it again and we would do more practice and the cycle would continue.
(As a side note, in my classroom I generally introduce students to manipulatives long before I ever use them in the classroom, and I often let students “play” with them during centre time. In doing this I find that students get their curiosity about the materials out of their systems long before we use them in math lessons and my lessons are much more productive because students already have some basic idea of what they are supposed to do with the manipulatives.)
I don’t ever really remember math being fun, though I was quite skilled in mathematics at a young age, and was even placed in an enrichment class for math and language in my elementary grades (a class which is interesting to note that was made up entirely of girls.) The enrichment class provided some opportunity for “problem solving” but in a very structured traditional sense, we were given problems, taught various strategies for solving them and then given a number of “new” problems to solve in the same fashion...I think this for many people would be what they think of when they refer to problem solving in math class, a unit taught in isolation to other mathematical skills and processes.
My primary/elementary teachers were wonderful; some could even be described as inspiring....in every way except for math class. When I look back I can think of a memorable lesson from every other subject in primary/elementary; class elections in social studies, bookmaking in language arts, interactive science experiments....I can’t however think of one math lesson that really excited me or stuck with me. Assessment in math....much the same.....nothing too exciting, pencil and paper tests and an assignment every now and then in the elementary grades.....I don't ever remember being asked to communicate in mathematics. Sure, I was asked the answer when the teacher would correct homework round-robin style every morning, but I was never asked how I got an answer (other than to work out the computation as an example) or why I thought my answer was right.
I can say with certainty however, that my worst memory from mathematics at this time would have to be in relation to homework, I am sure my teachers at this stage in my life believed firmly that if “some is good, more is better.” I remember having heaps and heaps of homework even in very low grades, often this homework was meant to reinforce or practice strategies taught at school, I often saw this as unfair to those of us who had mastered the skills in the lessons at school and I am sure it was often a struggle for my parents to get me to complete the work at home. (I wonder if my disdain for this homework caused me to detest it even more in higher grades when I certainly could have used the practice.) While I will admit to having given homework sheets to students (even in Grade 1) I have tried my best to give hands on tasks for homework to younger students, to provide them with a challenge and to allow them to make sense on their own of concepts that have been introduced in class. (I once sent students home during a measurement unit with a container of playdough, a small ball of wool that they were permitted to cut, a box of paperclips and a deck of playing cards. Students were instructed to find out who had the biggest feet/smallest feet in their house and order the feet from biggest to smallest. It was interesting to see how students completed this task, some used the paperclips to measure the length of the feet and could use numbers to describe the length of the feet, some rolled out playdough “snakes” and ordered them from longest to shortest, and others cut the string the appropriate lengths for each foot and then measured the strings with paperclips. Students thoroughly enjoyed this challenge, some even measured the feet on their pets, their chairs....and one student even measured the feet on the claw-footed tub!)
Perhaps my teachers didn’t feel comfortable with teaching math, although I can’t be certain I suspect that it may be partly true as I remember that there were times in Elementary grades in particular when we would often go days without a math class, and at the end of each year there were always units left uncovered, things that there simply “wasn’t time for”. (I recognize that there isn’t time for everything, and I myself have struggled with completing all of the curriculum outcomes. That being said, I would certainly not leave out an entire unit as I feel this would set students up for difficulties in future.)
In Junior High I continued to excel in math, although the battle with homework was one I continued to struggle with. In High School I took advanced math in grades 10 and 11 but dropped back to the academic program in grade 12. I remember my parents telling me that this was ok because I was more “artsy” anyway, and wouldn’t be as likely to need that math in the future. My sister was a year ahead of me in High School and I always felt that her success in math sometimes proved to be my detriment, with teachers asking how we could be so different, why I wasn’t as good as her in math, etc. I think that I heard this so many times that I eventually started to believe it and at times used it as an excuse for my poorer performance in the advanced program in grade 11.
After High School I was determined not to take any more math courses, I had done well in the grade 12 program, but still had never been very excited about math. When I began the Primary / Elementary Education pre-requisites I was required to take two math courses, not wanting to put myself through any more of what I was sure would be torture, than necessary, I decided to take Math 1050 and 1051. To my surprise I really enjoyed Math 1050, I liked that it was logic based and that it was something I could make sense of on my own. (I had always enjoyed Logic puzzles and was amazed that after so many years I was finally enjoying math.) The only other math course I took in University was Educ 3940, the math methods course.
Until a few years ago math wasn’t a big part of my life, sure, it’s useful, and I used it when I had to, but that was about it. Then I was assigned a grade 1 teaching position and I felt like I wasn’t as prepared for the position as I would like to be. At that point I made it my goal to participate in as much math professional development as possible. I began to work with other teachers as part of a Professional Learning Community developing ways to integrate children’s literature into our math lessons to provide our students with interesting lessons. I attended a week long workshop, “Math Their Way”, which allowed me to see how the effective use of manipulatives can influence math lessons in so many positive ways. I attended workshops on the new WNCP Math curriculum and its framework to help me understand where these changes in math instruction were coming from and where they will hopefully lead us.
My goal for increased professional competency has also led to an increased level of personal satisfaction in mathematics. I now feel much more comfortable in a math classroom as a teacher or student, I am hoping that this class will help me to help other students avoid some of the issues that I fell affected my own journey with mathematics.
From K-6 Math instruction in my classroom generally involved pencil and paper, the manipulatives were present in the classroom, but rarely made their way into lessons and when they did we were so in awe of these “blocks” that we were not permitted to “play” with at any other time, that little was ever really accomplished in terms of mathematics within these lessons. The teacher would use the chalkboard to do a few examples of whatever skill or strategy we were working on that day and then we would set to work on pages of practice. I remember getting new workbooks in Grade 2 as part of the MathQuest series, I also remember not doing any of the pages that involved cutting and pasting as these pages were "too messy and wasted too much time." In trying to find a way to describe the role of the teacher in my early math classes I came across the following Ten Roles for Math Teachers. I don't think any of these describe the math classrooms I experienced early in my schooling. I think it could best be described as teacher as demonstrator. The teacher demonstrated a skill, we were set about on our own to practice it, if we didn't understand it the teacher would demonstate it again and we would do more practice and the cycle would continue.
(As a side note, in my classroom I generally introduce students to manipulatives long before I ever use them in the classroom, and I often let students “play” with them during centre time. In doing this I find that students get their curiosity about the materials out of their systems long before we use them in math lessons and my lessons are much more productive because students already have some basic idea of what they are supposed to do with the manipulatives.)
I don’t ever really remember math being fun, though I was quite skilled in mathematics at a young age, and was even placed in an enrichment class for math and language in my elementary grades (a class which is interesting to note that was made up entirely of girls.) The enrichment class provided some opportunity for “problem solving” but in a very structured traditional sense, we were given problems, taught various strategies for solving them and then given a number of “new” problems to solve in the same fashion...I think this for many people would be what they think of when they refer to problem solving in math class, a unit taught in isolation to other mathematical skills and processes.
My primary/elementary teachers were wonderful; some could even be described as inspiring....in every way except for math class. When I look back I can think of a memorable lesson from every other subject in primary/elementary; class elections in social studies, bookmaking in language arts, interactive science experiments....I can’t however think of one math lesson that really excited me or stuck with me. Assessment in math....much the same.....nothing too exciting, pencil and paper tests and an assignment every now and then in the elementary grades.....I don't ever remember being asked to communicate in mathematics. Sure, I was asked the answer when the teacher would correct homework round-robin style every morning, but I was never asked how I got an answer (other than to work out the computation as an example) or why I thought my answer was right.
I can say with certainty however, that my worst memory from mathematics at this time would have to be in relation to homework, I am sure my teachers at this stage in my life believed firmly that if “some is good, more is better.” I remember having heaps and heaps of homework even in very low grades, often this homework was meant to reinforce or practice strategies taught at school, I often saw this as unfair to those of us who had mastered the skills in the lessons at school and I am sure it was often a struggle for my parents to get me to complete the work at home. (I wonder if my disdain for this homework caused me to detest it even more in higher grades when I certainly could have used the practice.) While I will admit to having given homework sheets to students (even in Grade 1) I have tried my best to give hands on tasks for homework to younger students, to provide them with a challenge and to allow them to make sense on their own of concepts that have been introduced in class. (I once sent students home during a measurement unit with a container of playdough, a small ball of wool that they were permitted to cut, a box of paperclips and a deck of playing cards. Students were instructed to find out who had the biggest feet/smallest feet in their house and order the feet from biggest to smallest. It was interesting to see how students completed this task, some used the paperclips to measure the length of the feet and could use numbers to describe the length of the feet, some rolled out playdough “snakes” and ordered them from longest to shortest, and others cut the string the appropriate lengths for each foot and then measured the strings with paperclips. Students thoroughly enjoyed this challenge, some even measured the feet on their pets, their chairs....and one student even measured the feet on the claw-footed tub!)
Perhaps my teachers didn’t feel comfortable with teaching math, although I can’t be certain I suspect that it may be partly true as I remember that there were times in Elementary grades in particular when we would often go days without a math class, and at the end of each year there were always units left uncovered, things that there simply “wasn’t time for”. (I recognize that there isn’t time for everything, and I myself have struggled with completing all of the curriculum outcomes. That being said, I would certainly not leave out an entire unit as I feel this would set students up for difficulties in future.)
In Junior High I continued to excel in math, although the battle with homework was one I continued to struggle with. In High School I took advanced math in grades 10 and 11 but dropped back to the academic program in grade 12. I remember my parents telling me that this was ok because I was more “artsy” anyway, and wouldn’t be as likely to need that math in the future. My sister was a year ahead of me in High School and I always felt that her success in math sometimes proved to be my detriment, with teachers asking how we could be so different, why I wasn’t as good as her in math, etc. I think that I heard this so many times that I eventually started to believe it and at times used it as an excuse for my poorer performance in the advanced program in grade 11.
After High School I was determined not to take any more math courses, I had done well in the grade 12 program, but still had never been very excited about math. When I began the Primary / Elementary Education pre-requisites I was required to take two math courses, not wanting to put myself through any more of what I was sure would be torture, than necessary, I decided to take Math 1050 and 1051. To my surprise I really enjoyed Math 1050, I liked that it was logic based and that it was something I could make sense of on my own. (I had always enjoyed Logic puzzles and was amazed that after so many years I was finally enjoying math.) The only other math course I took in University was Educ 3940, the math methods course.
Until a few years ago math wasn’t a big part of my life, sure, it’s useful, and I used it when I had to, but that was about it. Then I was assigned a grade 1 teaching position and I felt like I wasn’t as prepared for the position as I would like to be. At that point I made it my goal to participate in as much math professional development as possible. I began to work with other teachers as part of a Professional Learning Community developing ways to integrate children’s literature into our math lessons to provide our students with interesting lessons. I attended a week long workshop, “Math Their Way”, which allowed me to see how the effective use of manipulatives can influence math lessons in so many positive ways. I attended workshops on the new WNCP Math curriculum and its framework to help me understand where these changes in math instruction were coming from and where they will hopefully lead us.
My goal for increased professional competency has also led to an increased level of personal satisfaction in mathematics. I now feel much more comfortable in a math classroom as a teacher or student, I am hoping that this class will help me to help other students avoid some of the issues that I fell affected my own journey with mathematics.
Creativity, where does it come from, where does it go.....
The idea that creativity is as important as literacy, as put forth by Sir Ken Robinson is an idea that I found most intriguing.
Creativity is defined by www.dictionary.com as follows:
1. the state or quality of being creative.
2. the ability to transcend traditional ideas, rules, patterns, relationships, or the like, and to create meaningful new ideas, forms, methods, interpretations, etc.; originality, progressiveness, or imagination: the need for creativity in modern industry; creativity in the performing arts.
3. the process by which one utilizes creative ability: Extensive reading stimulated his creativity.
School Boards and Departments throughout the province, the country, and certainly even the world hire literacy consultants and directors, but I am fairly certain that there is not one position in education for a creativity director. But if we are in agreement with Robinson that this skill is in fact equally as important, how then can we foster this creativity in children?
Perhaps children come to school as their own creativity directors, as Robinson alludes, “kids have no fear of being wrong.” Over time however, most children develop a fear of being wrong and believe that a “mistake is the worst thing they can make.” This fear or lack of creativity comes from us as teachers if we fail to encourage our students to think outside the box, to make sense of learning on their own.
For example in problem solving in mathematics, children often find steps and strategies difficult to memorize and have trouble deciding which one to use when presented with a problem. According to Larry Buschman in Children Who Enjoy Problem Solving (TCM May 2003) “Young children want to solve problems, and their enjoyment of problem solving increases when children can solve problems in ways that make sense to them.” So, while these same children may not be able to follow the steps in a prescribed algorithm they may in fact be able to find the same answer to the same problem using a way that makes sense to them, a way that allows them to demonstrate creativity as opposed to memorization. Buschman says that he helps his students acquire the “habits, behaviours and disposition of a problem solver, such as patience, perseverance and a positive attitude.” In solving problems in this way, children aren’t faced with the same pressure to be right, they are able to use their mistakes to guide them towards their answer, this approach allows students to find many paths to the same answer, students are able to take much more pride in their work as they are now able to show what THEY know, as opposed to what they remembered. In adopting a less traditional approach to problem solving it is easy to see how creativity can be applied to mathematics, as students work on original ideas and solutions, as they work progressively towards solutions to problems, as they make sense of their own learning and form their own conclusions about mathematical rules, patterns and relationships.
Creativity, the lost skill.....definitely something interesting to think about.
Friday, September 11, 2009
It was great to meet everybody in the first class, this is my first semester in the Grad program, and this is my only on-campus class this semester, so it was nice to get a chance to connect with some "real" classmates (as opposed to my "virtual" classmates in my other courses.)
I'll try to take some time later today to collect and share my thoughts on the information/issues that were presented in last night's class.
Melanie
I'll try to take some time later today to collect and share my thoughts on the information/issues that were presented in last night's class.
Melanie
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