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?
Sunday, September 27, 2009
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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