As I mentioned in a previous entry, I have just recently begun homeschooling my daughter Kylie. Naturally, I have a very strong interest in her mathematical development. I want her to feel that all career options are available to her and the truth is that the lack of mathematical understanding is all too often the cause of career "doors" slamming shut. I want her to feel as if, should she choose, she could be a mathematician, a chemist, a physicist, an economist, an architect, a computer scientist, a pharmacist...you get the idea. Math plays an important role in many, many jobs. Those of you who have had me as an instructor know that I believe that if you are not running into situations that make you stop and really think about what you know and how you might use that information, then you aren't really learning in a way that is fostering intellectual development. So, we began the school year with what I would describe as "challenging, yet achievable" problems. Much to my surprise, Kylie was not very happy. Kylie has always loved math and enjoyed tackling difficult problems, so I had not expected this response. Part of the difficulty arose from the transition from school to homeschool. She was used to strategy shares when she could spend more time pondering other people's strategies than in sharing her own thinking. She is a bit of an introvert, so initially she found the idea of the "spotlight is just on me" to be overwhelming. Most of these issues have worked themselves out, but I have spent some time lately thinking about how to keep her love of math alive through the "tough spots."
I have just finished reading Jo Boaler's What's Math Got to Do With It? ( Subtitle: Helping Children Learn to Love Their Least Favorite Subject--And Why It's Important for America). In this book she mentioned that many famous mathematicians did not develop their love for the subject in school, but rather from someone who shared mathematical puzzles with them. The mathematical puzzles were intriguing and pulled them in to mathematics in joyful ways. So, I introduced Kylie first to the container problem (see sample puzzles at the end of this entry). She was riveted, frustrated, and ultimately thrillingly victorious! I do realize that math instruction cannot consist of just a steady diet of puzzles. However, I have come to see the need for them to play a role in the big picture of mathematical instruction. I don't pretend to have the "perfect" plan for approaching math, and I'm sure that this will evolve, but currently we have a 3-phase math time. Initially, we have a series of mathematical exercises that have been specially chosen to highlight certain mathematical relationships, suggest efficient, sense-based strategies, and/or see the use of particular mathematical models. If you are a former student, you may recognize this as the "computational minilesson" that Fosnot/Dolk talked about in Young Mathematicians at Work (by the way, if you liked those books, did you know that they now have supplemental curriculum available? Check it out at http://www.heinemann.com/ ). Following this time of strategy development, we do the concept-development phase of instruction. For this portion, I am using the Japanese curriculum that I mentioned in an earlier entry as a general guide. I do supplement it depending on Kylie's needs. The final portion of the class is usually choice time. I have given her a selection of mathematical puzzles and extended problems to explore. She chooses which ones she wants to do, and approaches them in any order, depending on how they intrigue her. Adding this component has had a hugely positive impact on her attitude. I also give her a "menu" of open-ended problems related to our current topic part way into the unit. She has to have these done by the end of the unit, but this still allows plenty of time for her to explore other problems.
These three phases allow Kylie to: 1) develop her strategies and grow in her understanding of number relationships and mental math; 2)move her forward into newer concepts in a way that builds on her previous schema, and; 3) allows her to see herself as a true mathematician tackling intriguing problems.
Container Puzzle:
Given a five-liter container, a three-liter container, and an unlimited supply of water, how do you measure out four liters exactly?
The Rabbit Puzzle: (also taken from Jo Boaler's book)
A rabbit falls into a dry well thirty meters deep. Since being at the bottom of the well was not her original plan, she decides to climb out. When she attempts to do so, she finds that after going up three meters (and this is the sad part) she slips back two. Frustrated, she stops where she is for the day and resumes her efforts the following morning--with the same results. How many days does it take her to get out of the well?
The Chessboard Problem:
How many squares are on a standard 8 x 8 chessboard? (Keep in mind that this is referring to squares of all sizes). How can you know for sure if you have found all of the possibilities? Can you generalize a method for finding out the number of squares of any chessboard of with different edge lengths? (For Kylie, I printed off sheets of empty chessboard clipart that she can use to write on, if she wishes. This is the problem she is currently working on. I haven't seen her work. She prefers to do her thinking on these problems on her own as much as possible.)
For some online mathematical puzzles, check out these links. The first one is interactive:
http://www.math.com/students/puzzles/puzzleapps.html
http://www.jimloy.com/puzz/puzz.htm
http://thinks.com/puzzles/loyd/loyd.htm
Thursday, September 25, 2008
What does it mean to be a professional?
Teaching is considered to be a profession. We certainly expect to be treated like professionals. I propose that the varied uses of that word have obscured the true expectations we should hold ourselves to and the rights that we should advocate for as professionals. Some of the problem can be blamed on one of the most common uses of the word--to "act professionally." Certainly, we are (and should be) expected to dress appropriately, speak appropriately, and interact with others appropriately. But this is not what makes us a professional. If a receptionist is chewing gum, blowing bubbles behind her desk, and texting her friends before waiting on me, then I may say that she is not acting very professional. However, being a receptionist is not a professional job. So clearly, acting professionally and being a professional are two separate things. To explore the meaning of the word "professional" as it applies to teaching, I will use a metaphor to compare it to being a doctor--a career for which we have high expectations of professionals.
I suppose it is possible to be a "technical" doctor--one who is more comfortable following prescribed protocols than in fostering patient's overall health. Such a "technical" doctor might listen to patients only long enough to do a "protocol match", "Oh, this is the problem, prescribe X." She might treat isolated symptoms rather than thinking and problem solving through the complexities of multiple symptoms. She might see all problems as variations of issues that were previously learned about in medical school; therefore, there would be no perceived need for on-going learning and research. She may not see any urgency in keeping up with the latest research. If her prescribed protocols don't work and the patient continues to experience problems, she may become frustrated and imagine that the patient is probably just depressed or a hypochondriac (stupid or not trying). I call this a "technical" doctor, because this is not what should be expected of a professional. Rather, this doctor is a technician--someone who follows procedures without demonstrating deep professional understandings. How does this look in a teaching setting? Technical teachers follow prescribed protocols when they follow curriculum blindly without considering carefully the needs of their students. Who is ready for this lesson? Is it a rich enough lesson that it will offer some potential for growth in all of my students or will it really only address the needs of the "middle"? The isolated symptoms that are treated are the isolated skills that are addressed. People's health (and their minds) are complex. The symptoms do not always (or even usually) stand in isolation. We cannot assume that treating all the isolated bits will address the problems of the "whole".
Professional doctors see patients as individuals with unique situations, while also being able to relate their profile to a database of known information about patient health. One of the things that especially drew me to Cognitively Guided Instruction was the combination of these two features--a rich database of children's cognitive development in mathematics to help guide your instructional decision-making as you listen to and carefully observe your individual student(s). The professional doctor recognizes that diseases may not fit a "one size fits all" description in the manner that it manifests itself. She listens to all symptoms, asks questions to find out more about the internal workings of their bodies, generates multiple potentially valid hypotheses about the condition(s), and (rather than following a prescribed protocol automatically) considers carefully whether such protocols will be effective in this situation, whether adaptations need to be made, or whether totally new protocols need to be developed. The professional doctor does not assume that apparent health means that there is no need for action, just as a professional teacher does not assume that gifted and high-ability students can be left to their own devices. She is intrigued with "mystery" cases and works hard to help these patients progress. "Working hard" doesn't mean trying out treatments willy-nilly, but rather doing individual research, consulting with colleagues, and finding out more information about the patient and listening more closely to them. The longer she practices, the more aware she is of what she doesn't know, and the greater her need to develop herself professionally. In the teaching world, this need to develop oneself professionally goes far beyond attending inservices and even beyond going back to earn a master's degree. It involves a deep personal commitment to be involved in professional organizations, read professional journals and books, and to seek ways every year to improve our practice in ways that distinctly benefit our students' intellectual development.
How many teachers do you know who truly are professionals by the above standard? I have been privileged to meet some amazing teachers who do fit this description. Seeing them in practice only highlights what a rare phenomenon this really is (I believe the majority of teachers are wonderful, caring individuals. As you read further, you will see that this is not about teacher-bashing. Rather, I am decrying the fact that teachers are not often given the opportunity to be all that they are capable of being) . I find myself wondering why there are so few of these highly professional teachers. Teachers are not paid as well as doctors, but most teachers I know did not go into it for the pay, so I don't buy that argument. I have two hypotheses that seem quite reasonable to me, although I suspect that it is one of those complex problems with many influences. One likely reason is that the teachers are not only not expected to behave like this; they are often treated as if they should NOT be truly professional. That sounds like a radical statement, but it usually manifests itself in a couple of familiar ways. An administrator may tell a teacher that she has to follow a particular curriculum to the letter. This is like telling a teacher not to consider the needs of their students in making instructional decisions. Teachers may also feel pressure from other teachers, particularly those with more status, to conform to "the way we do it here." If there is no overt encouragement for teachers to be researching and applying best practice, then the environment is anti-professional (or a more favorable interpretation is that they are NOT pro-professional). Another likely reason follows from this one. Because many schools are not actively encouraging best practice, it turns out that many teachers have not seen it in practice. It is difficult (although not impossible) to put into practice what you haven't experienced personally. Sadly, many teachers have spent long enough in this professionally-deprived environment that they have forgotten that they are capable of so much more.
So what advice might one give a teacher who truly aspires to be a "professional"? First and foremost is to read, read, read about best practice. It is easy to say that you don't have time. Students laugh, but my advice for professional reading is to: 1) find a publisher who you know is sound when it comes to best practice (http://www.heinemann.com/ is one of my favorites); 2) keep a wishlist handy--offer it to family members for Christmas and birthdays or use it if you get an Amazon gift card; 3) keep your professional books in the following places--the headboard of your bed, the back of your toilet, and in your car. If you have insomnia, read. If you like to read in the bathroom, you're in business! If you have a doctor's appointment or an oil change or you are killing time waiting for your daughter's dance lesson to end, you have something decent to read; 4) push yourself to try at least one new instructional approach each year; ideally one that you have targeted to address a particular need you have noticed in your students. Expect that things will not go smoothly right away. Plan to spend some time problem solving the difficulties that naturally arise when you try something unfamiliar.
The second piece of advice I have is for you to believe in yourself and your ability to change. You will very likely find yourself in a place where change is not being encouraged. Believe in yourself, and (just as importantly) know why you believe what you do and be prepared to defend it in polite, well-reasoned, well-supported ways (now you see the reason for that reading above). Of course, teaching this way is hard work, but it is very rewarding in terms of fulfillment and self-efficacy. If you do find resistance, consider who your support system is that will give you the courage to keep going. If there is no one nearby or in your school, consider a listserv. Speaking from personal experience, I can tell you that dramatic improvement in your professional practice is possible, even in work environments where your colleagues would just as soon that your teaching approach failed spectacularly!
The third (and most important) piece of advice is to listen to your students. What do you see as their needs? Think deeply about this and identify the gaps in your own understanding. Those "gaps" will be the best impetus ever to drive your desire to be a professional in the truest sense of the word. I wish you all the best in this endeavor.
I suppose it is possible to be a "technical" doctor--one who is more comfortable following prescribed protocols than in fostering patient's overall health. Such a "technical" doctor might listen to patients only long enough to do a "protocol match", "Oh, this is the problem, prescribe X." She might treat isolated symptoms rather than thinking and problem solving through the complexities of multiple symptoms. She might see all problems as variations of issues that were previously learned about in medical school; therefore, there would be no perceived need for on-going learning and research. She may not see any urgency in keeping up with the latest research. If her prescribed protocols don't work and the patient continues to experience problems, she may become frustrated and imagine that the patient is probably just depressed or a hypochondriac (stupid or not trying). I call this a "technical" doctor, because this is not what should be expected of a professional. Rather, this doctor is a technician--someone who follows procedures without demonstrating deep professional understandings. How does this look in a teaching setting? Technical teachers follow prescribed protocols when they follow curriculum blindly without considering carefully the needs of their students. Who is ready for this lesson? Is it a rich enough lesson that it will offer some potential for growth in all of my students or will it really only address the needs of the "middle"? The isolated symptoms that are treated are the isolated skills that are addressed. People's health (and their minds) are complex. The symptoms do not always (or even usually) stand in isolation. We cannot assume that treating all the isolated bits will address the problems of the "whole".
Professional doctors see patients as individuals with unique situations, while also being able to relate their profile to a database of known information about patient health. One of the things that especially drew me to Cognitively Guided Instruction was the combination of these two features--a rich database of children's cognitive development in mathematics to help guide your instructional decision-making as you listen to and carefully observe your individual student(s). The professional doctor recognizes that diseases may not fit a "one size fits all" description in the manner that it manifests itself. She listens to all symptoms, asks questions to find out more about the internal workings of their bodies, generates multiple potentially valid hypotheses about the condition(s), and (rather than following a prescribed protocol automatically) considers carefully whether such protocols will be effective in this situation, whether adaptations need to be made, or whether totally new protocols need to be developed. The professional doctor does not assume that apparent health means that there is no need for action, just as a professional teacher does not assume that gifted and high-ability students can be left to their own devices. She is intrigued with "mystery" cases and works hard to help these patients progress. "Working hard" doesn't mean trying out treatments willy-nilly, but rather doing individual research, consulting with colleagues, and finding out more information about the patient and listening more closely to them. The longer she practices, the more aware she is of what she doesn't know, and the greater her need to develop herself professionally. In the teaching world, this need to develop oneself professionally goes far beyond attending inservices and even beyond going back to earn a master's degree. It involves a deep personal commitment to be involved in professional organizations, read professional journals and books, and to seek ways every year to improve our practice in ways that distinctly benefit our students' intellectual development.
How many teachers do you know who truly are professionals by the above standard? I have been privileged to meet some amazing teachers who do fit this description. Seeing them in practice only highlights what a rare phenomenon this really is (I believe the majority of teachers are wonderful, caring individuals. As you read further, you will see that this is not about teacher-bashing. Rather, I am decrying the fact that teachers are not often given the opportunity to be all that they are capable of being) . I find myself wondering why there are so few of these highly professional teachers. Teachers are not paid as well as doctors, but most teachers I know did not go into it for the pay, so I don't buy that argument. I have two hypotheses that seem quite reasonable to me, although I suspect that it is one of those complex problems with many influences. One likely reason is that the teachers are not only not expected to behave like this; they are often treated as if they should NOT be truly professional. That sounds like a radical statement, but it usually manifests itself in a couple of familiar ways. An administrator may tell a teacher that she has to follow a particular curriculum to the letter. This is like telling a teacher not to consider the needs of their students in making instructional decisions. Teachers may also feel pressure from other teachers, particularly those with more status, to conform to "the way we do it here." If there is no overt encouragement for teachers to be researching and applying best practice, then the environment is anti-professional (or a more favorable interpretation is that they are NOT pro-professional). Another likely reason follows from this one. Because many schools are not actively encouraging best practice, it turns out that many teachers have not seen it in practice. It is difficult (although not impossible) to put into practice what you haven't experienced personally. Sadly, many teachers have spent long enough in this professionally-deprived environment that they have forgotten that they are capable of so much more.
So what advice might one give a teacher who truly aspires to be a "professional"? First and foremost is to read, read, read about best practice. It is easy to say that you don't have time. Students laugh, but my advice for professional reading is to: 1) find a publisher who you know is sound when it comes to best practice (http://www.heinemann.com/ is one of my favorites); 2) keep a wishlist handy--offer it to family members for Christmas and birthdays or use it if you get an Amazon gift card; 3) keep your professional books in the following places--the headboard of your bed, the back of your toilet, and in your car. If you have insomnia, read. If you like to read in the bathroom, you're in business! If you have a doctor's appointment or an oil change or you are killing time waiting for your daughter's dance lesson to end, you have something decent to read; 4) push yourself to try at least one new instructional approach each year; ideally one that you have targeted to address a particular need you have noticed in your students. Expect that things will not go smoothly right away. Plan to spend some time problem solving the difficulties that naturally arise when you try something unfamiliar.
The second piece of advice I have is for you to believe in yourself and your ability to change. You will very likely find yourself in a place where change is not being encouraged. Believe in yourself, and (just as importantly) know why you believe what you do and be prepared to defend it in polite, well-reasoned, well-supported ways (now you see the reason for that reading above). Of course, teaching this way is hard work, but it is very rewarding in terms of fulfillment and self-efficacy. If you do find resistance, consider who your support system is that will give you the courage to keep going. If there is no one nearby or in your school, consider a listserv. Speaking from personal experience, I can tell you that dramatic improvement in your professional practice is possible, even in work environments where your colleagues would just as soon that your teaching approach failed spectacularly!
The third (and most important) piece of advice is to listen to your students. What do you see as their needs? Think deeply about this and identify the gaps in your own understanding. Those "gaps" will be the best impetus ever to drive your desire to be a professional in the truest sense of the word. I wish you all the best in this endeavor.
Wednesday, September 3, 2008
Subliminal Messages Sent to Children About Mathematics
My oldest daughter is entering 5th grade. For the first time (at least formally) we are homeschooling her. Given that math education is one of my fields, I have quite a large collection of resources and curricular materials to draw from. Actually, making a decision was quite challenging, because there were many good options. As you may notice, one of the blogs I enjoy reading is that of Tad Watanabe. He has advocated for a curriculum that is more coherent, and which builds conceptual understanding as well as procedural understanding. He is one of the supporters of Global Education Resources (http://www.globaledresources.com/) which sells English translation of Japanese textbooks. They are fairly reasonable, so I decided to order the 5th grade textbook. The first thing I noticed is that the year's curriculum comes as two fairly thin paperbacks about the size of a 5 x 7 picture on the front. There is no teacher's manual, at least not of the kind that we are accustomed to seeing here in the U.S. that lays out some prescribed approach (without knowing the needs of the students). Instead, you are advised to purchase the Teachers' Guidelines document, which is on a CD. It lays out specific, grade level target objectives and provides guidelines, such as ensuring that students have opportunities to model relationships and operations with concrete objects before beginning to introduce algorithms. It is a very useful document for anyone wishing to teach a child mathematics, regardless of their country of origin.
When I opened the front cover of the text, I was immediately taken with the introductory paragraph directed to students. It seemed to reflect vastly different attitudes and ideas about mathematics. It really drew a contrast in my mind between the kinds of attitudes that are fostered in Japan and the subliminal messages that American children receive. Shown below is that first paragraph from the Japanese text in bold print, along with the italicized comments, indicating the contrasting subliminal message I think U.S. children pick up.
There are many children who think that studying mathematics is interesting. Math is boring. Very few children like it, unless they are "math geeks." That is probably because, in mathematics, you can usually find the answer if you think persistently and diligently. Some people (maybe many) just can't do math. Perhaps they didn't inherit the math gene. It is like climbing up stairs one by one, although it may take time. You either know how to do it or you don't.
In mathematics, there is not only one way to find an answer. You have to do it exactly as the teacher shows you or it will be wrong. If you figure out another way, that is like cheating. So it is important to listen to your friends' ideas and discuss the ideas with them. You shouldn't discuss your ideas and answers with anyone else, or else you could get in trouble for cheating.
Let's work hard and do challenging mathematics together. If you don't know how to do something, ask for help from the teacher right away. They are supposed to "make it easy".
These subliminal messages play a significant role in the "mathematical health" of our nation. One study that I read on problem solving contrasted the differences between how Japanese first graders responded to a challenging problem compared to American first graders. The Japanese students worked on the same problem for 45 minutes, which is when the teacher concluded the lesson. The American students all gave up on the problem within 3 minutes. Clearly, those Japanese children believed that they could make sense and solve that problem, if they were just persistent. How do you suppose they developed that belief? Being told that persistence is important is a starting point, but I suspect those students had had plenty of opportunity to exercise that persistence without a teacher coming in at the first sign of confusion to "save the day."
What subliminal messages did you receive as a child about mathematics? How might you "break the cycle" of negative attitudes and foster a healthier, more productive, problem-solving attitude? These are questions well worth pondering.
When I opened the front cover of the text, I was immediately taken with the introductory paragraph directed to students. It seemed to reflect vastly different attitudes and ideas about mathematics. It really drew a contrast in my mind between the kinds of attitudes that are fostered in Japan and the subliminal messages that American children receive. Shown below is that first paragraph from the Japanese text in bold print, along with the italicized comments, indicating the contrasting subliminal message I think U.S. children pick up.
There are many children who think that studying mathematics is interesting. Math is boring. Very few children like it, unless they are "math geeks." That is probably because, in mathematics, you can usually find the answer if you think persistently and diligently. Some people (maybe many) just can't do math. Perhaps they didn't inherit the math gene. It is like climbing up stairs one by one, although it may take time. You either know how to do it or you don't.
In mathematics, there is not only one way to find an answer. You have to do it exactly as the teacher shows you or it will be wrong. If you figure out another way, that is like cheating. So it is important to listen to your friends' ideas and discuss the ideas with them. You shouldn't discuss your ideas and answers with anyone else, or else you could get in trouble for cheating.
Let's work hard and do challenging mathematics together. If you don't know how to do something, ask for help from the teacher right away. They are supposed to "make it easy".
These subliminal messages play a significant role in the "mathematical health" of our nation. One study that I read on problem solving contrasted the differences between how Japanese first graders responded to a challenging problem compared to American first graders. The Japanese students worked on the same problem for 45 minutes, which is when the teacher concluded the lesson. The American students all gave up on the problem within 3 minutes. Clearly, those Japanese children believed that they could make sense and solve that problem, if they were just persistent. How do you suppose they developed that belief? Being told that persistence is important is a starting point, but I suspect those students had had plenty of opportunity to exercise that persistence without a teacher coming in at the first sign of confusion to "save the day."
What subliminal messages did you receive as a child about mathematics? How might you "break the cycle" of negative attitudes and foster a healthier, more productive, problem-solving attitude? These are questions well worth pondering.
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