I am teaching a workshop in San Francisco, June 20-21, on "reimagining high school math".
Support materials are on the workshop participants' Web site.
Participants: please use the comments below to share ideas, resources, and questions.
--Henri
Friday, June 14, 2013
Thursday, June 13, 2013
Visual Algebra: Sharing
I am teaching a Visual Algebra workshop in San Francisco, June 17-19.
Support materials are on the workshop participants' Web site.
Over the past couple of decades, there has been a trend to teach algebra to younger and younger students. That could actually be a good thing. But unfortunately, that has been interpreted as teaching the traditional Algebra 1 course to younger and younger students. That is not a good thing, as that course was problematic: too abstract, too authoritarian, too boring, and seemingly irrelevant. I have written much on this topic -- you can get links to my various articles here.
The purpose of the workshop is to offer some visual approaches to algebra, that make the course more accessible, deeper, and more interesting.
Participants: please use the comments below to share ideas, resources, and questions.
--Henri
Support materials are on the workshop participants' Web site.
Over the past couple of decades, there has been a trend to teach algebra to younger and younger students. That could actually be a good thing. But unfortunately, that has been interpreted as teaching the traditional Algebra 1 course to younger and younger students. That is not a good thing, as that course was problematic: too abstract, too authoritarian, too boring, and seemingly irrelevant. I have written much on this topic -- you can get links to my various articles here.
The purpose of the workshop is to offer some visual approaches to algebra, that make the course more accessible, deeper, and more interesting.
Participants: please use the comments below to share ideas, resources, and questions.
--Henri
Sunday, June 9, 2013
Transformational Geometry, cont'd.
As I mentioned in a recent post, I will be one of the presenters at the Bay Area Math Project's summer workshop on Transformational Geometry. As part of preparing for this, I went through my notes, and compiled a sort of syllabus of the relevant lessons from my Space course. Symmetry and transformations are the backbone of the course, and will grow in importance in the mainstream of high school math education if the Common Core State Standards take hold. If you will be trying to implement the CCSS, you may find my syllabus to be a useful reference. It is posted here, on the site I am building for my 2013 summer workshops.
--Henri
--Henri
Friday, June 7, 2013
Embracing Contraries
In 1983, Peter Elbow, a professor at Stony Brook University, wrote a profound article about teaching. It was titled "Embracing Contraries in the Teaching Process" and was published in College English, Volume 45, Number 4.
The article was very important for me. It helped me clarify my thinking about education — even though Elbow is a teacher of English and I teach math. I will not attempt to summarize his whole argument: I encourage you to read it on line here. (This site requires registration, but it is free. Just don't download thousands of articles from there, if you want to avoid being prosecuted like Aaron Swartz.) A lower-quality copy of the article is available here (no registration needed.)
Near the opening of the article, Elbow writes:
Choosing only one of these "contraries" is terribly limiting. So-called "soft" teachers prioritize their commitment to the students. So-called "hard" teachers prioritize their commitment to the discipline. Good teaching requires not a compromise between the two, but an ability to do both at different times. Elbow suggests that at the start of a course you should be "hard" and make clear your criteria for grading. After that, you should be "soft" and be 100% on the student's side to help them achieve the goals you laid out. And finally, at the end of the course, you should be "hard" again, and give them the grade they deserve. Of course, this is oversimplified, and you'll need to constantly navigate along this axis -- sometimes emphasizing one commitment, sometimes the other.
Elbow is right: grading is in large part about commitment to the discipline (though this is sometimes tempered by an attempt to make the grades reward student obedience, which is perhaps what Elbow is getting at by mentioning "society".) I just retired after 32 years in the Math Department at the Urban School of San Francisco. The school's grading policy to some extent embodies Elbow's advice: students get abundant feedback during the course, but not grades. They do get one grade for the course, which they only find out at the end. This profoundly counter-cultural policy allows the focus of the teacher-student relationship to be student learning, and helps teacher resist the alas common debasing of the educational enterprise to a crass negotiation over points.
Of course grades are the currency in most schools, but that does not invalidate Elbow's point: you become most effective if you learn to embrace contraries and navigate between them. This is key to almost every facet of teaching. See the worksheet I developed for teachers to reflect upon their practice. In it, I listed as many opposites as I could think of. Let me know if you can think of more!
--Henri
The article was very important for me. It helped me clarify my thinking about education — even though Elbow is a teacher of English and I teach math. I will not attempt to summarize his whole argument: I encourage you to read it on line here. (This site requires registration, but it is free. Just don't download thousands of articles from there, if you want to avoid being prosecuted like Aaron Swartz.) A lower-quality copy of the article is available here (no registration needed.)
Near the opening of the article, Elbow writes:
... the two conflicting mentalities needed for good teaching stem from the two conflicting obligations inherent in the job: we have an obligation to students but we also have an obligation to knowledge and society. Surely we are incomplete as teachers if we are committed only to what we are teaching but not to our students, or only to our students but not to what we are teaching, or halfhearted in our commitment to both.To fully grasp the significance of this, you have to see that these two obligations are often in direct opposition to each other. For example: my commitment to mathematics requires me to read a student-written proof with a relentlessly critical eye, making sure the logic is tight and the writing clear. But my commitment to the student requires me to be encouraging and supportive, to look for whatever germs of good thinking are there even if the proof is flawed, and to find ways to build on those. These are nearly opposite, and yet I need to be totally committed to both.
Choosing only one of these "contraries" is terribly limiting. So-called "soft" teachers prioritize their commitment to the students. So-called "hard" teachers prioritize their commitment to the discipline. Good teaching requires not a compromise between the two, but an ability to do both at different times. Elbow suggests that at the start of a course you should be "hard" and make clear your criteria for grading. After that, you should be "soft" and be 100% on the student's side to help them achieve the goals you laid out. And finally, at the end of the course, you should be "hard" again, and give them the grade they deserve. Of course, this is oversimplified, and you'll need to constantly navigate along this axis -- sometimes emphasizing one commitment, sometimes the other.
Elbow is right: grading is in large part about commitment to the discipline (though this is sometimes tempered by an attempt to make the grades reward student obedience, which is perhaps what Elbow is getting at by mentioning "society".) I just retired after 32 years in the Math Department at the Urban School of San Francisco. The school's grading policy to some extent embodies Elbow's advice: students get abundant feedback during the course, but not grades. They do get one grade for the course, which they only find out at the end. This profoundly counter-cultural policy allows the focus of the teacher-student relationship to be student learning, and helps teacher resist the alas common debasing of the educational enterprise to a crass negotiation over points.
Of course grades are the currency in most schools, but that does not invalidate Elbow's point: you become most effective if you learn to embrace contraries and navigate between them. This is key to almost every facet of teaching. See the worksheet I developed for teachers to reflect upon their practice. In it, I listed as many opposites as I could think of. Let me know if you can think of more!
--Henri
Wednesday, May 29, 2013
Intelligent sequencing vs. external mandates
Some time ago, Mike Thayer posted a comparison of Algebra 1 and Geometry as they are experienced in the classroom. He concluded that since geometry is so much more real to students, and lends itself to interesting connections, perhaps it should be taught first.
In response, I suggested that moving specific topics up and down the high school sequence is a more flexible tool than moving entire courses, and I cited my experience at the Urban School, where we had some success with this approach:
- Math 1: basic intro to functions and graphs (linear, quadratic, exponential), a manipulatives-based approach to symbol manipulation, and "real world" (including geometric) contexts
- Math 2: largely geometry, some programming, some algebra (systems of equations, working with radicals), some basic trig
- Math 3: quadratics in depth, iterating linear functions, plus many standard Algebra 2 topics
Note that this is largely about postponing traditional end-of-the-book Algebra 1 topics until the students are more ready for them. (There is more info about that curriculum here.)
In response, Mike pointed out that such flexibility is not available to public school teachers, as the curriculum they are supposed to teach is mandated externally.
Since my career has been entirely in private schools, I know very little about the politics of how the public school curriculum is set. Nevertheless, I will make two suggestions.
The first is to use the coming of the Common Core State Standards as an opportunity to ask for a reevaluation of the sequencing of topics. The CCSS do not mandate any particular sequence in high school. They can be implemented through integrated courses, or in the traditional sequence, or even in some other sequence altogether. So for example, some version of the sequence suggested above would be legitimate.
Moreover, the CCSS require substantial shifts in emphasis. In algebra, modeling and families of functions take a bigger role, and manipulation of symbols and "simplifying" are demoted. In geometry, transformations become foundational, and congruence / similarity postulates follow. Those changes are tremendous opportunities to create a program that is at the same time more accessible, more interesting, and mathematically deeper. I hope that teachers who want these outcomes will take the lead in the coming curriculum discussions, because inertia reigns in the teaching of high school math, and there will be tremendous resistance.
My second thought is that even within a problematic mandated curriculum, there is always some wiggle room. One can and must prioritize, because superficial coverage of too many topics is in no one's interest. It does not even adequately prepare students for the standardized tests that are supposed to reflect that content. Much more effective is to choose what to emphasize, and give more time and multifaceted teaching to those topics. Yes, that means spending less time on other things, but on balance, that approach is sure to yield better results in understanding, motivation, and —very likely— test scores.
--Henri
In response, I suggested that moving specific topics up and down the high school sequence is a more flexible tool than moving entire courses, and I cited my experience at the Urban School, where we had some success with this approach:
- Math 1: basic intro to functions and graphs (linear, quadratic, exponential), a manipulatives-based approach to symbol manipulation, and "real world" (including geometric) contexts
- Math 2: largely geometry, some programming, some algebra (systems of equations, working with radicals), some basic trig
- Math 3: quadratics in depth, iterating linear functions, plus many standard Algebra 2 topics
Note that this is largely about postponing traditional end-of-the-book Algebra 1 topics until the students are more ready for them. (There is more info about that curriculum here.)
In response, Mike pointed out that such flexibility is not available to public school teachers, as the curriculum they are supposed to teach is mandated externally.
Since my career has been entirely in private schools, I know very little about the politics of how the public school curriculum is set. Nevertheless, I will make two suggestions.
The first is to use the coming of the Common Core State Standards as an opportunity to ask for a reevaluation of the sequencing of topics. The CCSS do not mandate any particular sequence in high school. They can be implemented through integrated courses, or in the traditional sequence, or even in some other sequence altogether. So for example, some version of the sequence suggested above would be legitimate.
Moreover, the CCSS require substantial shifts in emphasis. In algebra, modeling and families of functions take a bigger role, and manipulation of symbols and "simplifying" are demoted. In geometry, transformations become foundational, and congruence / similarity postulates follow. Those changes are tremendous opportunities to create a program that is at the same time more accessible, more interesting, and mathematically deeper. I hope that teachers who want these outcomes will take the lead in the coming curriculum discussions, because inertia reigns in the teaching of high school math, and there will be tremendous resistance.
My second thought is that even within a problematic mandated curriculum, there is always some wiggle room. One can and must prioritize, because superficial coverage of too many topics is in no one's interest. It does not even adequately prepare students for the standardized tests that are supposed to reflect that content. Much more effective is to choose what to emphasize, and give more time and multifaceted teaching to those topics. Yes, that means spending less time on other things, but on balance, that approach is sure to yield better results in understanding, motivation, and —very likely— test scores.
--Henri
Tuesday, May 14, 2013
Symmetry in Spain
My wife had a conference in Spain, so we built a vacation around that. It turns out that Spain is a fun place to visit: great art, great food, and well, great math. For example, you can buy a book about the golden ratio from a news kiosk on the street:
Unlike us as a culture, they are not afraid of 0 or negative numbers. Elevators indicate the ground floor as 0, and floors beneath as negative numbers:
Given that I have discussed symmetry in Islamic art for a couple of decades in my Space class, I wanted to make sure we would visit the Alhambra, and we did, towards the end of our trip.
The Alhambra is an architectural marvel, and Spain's most visited site. (More info on Wikipedia.) For a math teacher, it is remarkable in the variety of the symmetric patterns in the tile designs and stonework. Apparently, a young M.C. Escher visited the Alhambra and was inspired to explore tilings and symmetry in his artwork.
Oh, and speaking of symmetry in art, the Reina Sofia museum in Madrid had an exhibit of Latin American geometric abstract art from the middle of the 20th century. Two examples:
Like these two, many works in that exhibit had two-fold rotational symmetry. See more examples on the exhibit's own Web site: Concrete Invention.
Knowing that we would end up at the Alhambra, I took pictures of symmetric designs whenever I thought of it during the trip. You can see more here, culminating with my Alhambra photos. (Click on the first photo to see the photos one at a time -- a few have captions.)
--Henri
Given that I have discussed symmetry in Islamic art for a couple of decades in my Space class, I wanted to make sure we would visit the Alhambra, and we did, towards the end of our trip.
The Alhambra is an architectural marvel, and Spain's most visited site. (More info on Wikipedia.) For a math teacher, it is remarkable in the variety of the symmetric patterns in the tile designs and stonework. Apparently, a young M.C. Escher visited the Alhambra and was inspired to explore tilings and symmetry in his artwork.
Oh, and speaking of symmetry in art, the Reina Sofia museum in Madrid had an exhibit of Latin American geometric abstract art from the middle of the 20th century. Two examples:
Like these two, many works in that exhibit had two-fold rotational symmetry. See more examples on the exhibit's own Web site: Concrete Invention.
Knowing that we would end up at the Alhambra, I took pictures of symmetric designs whenever I thought of it during the trip. You can see more here, culminating with my Alhambra photos. (Click on the first photo to see the photos one at a time -- a few have captions.)
--Henri
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