Division by zero causes black holes
5/23/09
5/21/09
Photos are worth thousands of words
On a cold, windy yet sunny SF day (the norm), my 7th graders worked out the best possible outcomes for their water balloon bungees. After testing how far a balloon would fall with two, four and six rubber bands, they created a graph and measured the height of the middle school balcony using the concept of similar triangles we had studied months ago. They made a prediction of the number of rubber bands, tied them on, and we all went out to see the results. Two groups got their balloons within a few centimeters of the ground, while other, more conservative groups, did not put quite enough rubber bands out of fear that their data wasn't accurate enough. One group lost their balloon bungee jumper to a concrete finale.
A very fun day and a great way to learn about the power of mathematical modeling.
See photos on my other blog: http://inside.sfschool.org/users/gkenyon/
Why Me?
Why Do I Have to Take Algebra?
Students frequently question the usefullness of algebra, and express various objections to "having" to take an algebra class. But do these objections stand up under scrutiny?
"I don't need algebra, because I'm not going to college": There was a time not so long ago when children in middle schools were assigned to "tracks" according to what "everybody knew" each child would "need". (This tracking was why middle schools were invented in the first place.) Educational "experts" presumed to "know" what the various children "needed", based on culturally-based (but unjustified) presumptions. The educators then locked children into "appropriate" tracks, thereby locking many children out of college before they'd even begun high school.
It might have been assumed, for instance, that Shaniqwa would be pregnant by the time she was fourteen, Jamal would be in prison, José would grow up to be a pool-boy, and Maria would be a maid. So these students would have been assigned to something like "consumer math": low-level math that was presumed to be "useful" for "that sort". Blonde, blue-eyed Tiffany might have been expected to marry well after a short and trivial "career", so she'd have been assigned to bookkeeping. Only Eustace James Whittington III would have had any chance of attending college, so only he would have been steered into the algebra class.
I would hate to see a return to those days, and I can't understand why any student would volunteer to put himself into the position that used to be forced on many women and minorities. Even if college isn't currently in your plans, please don't under-value yourself by classifying yourself as "that sort" by thinking that you could never use algebra. Don't diminish your potential by rejecting mathematics.
"Having to take algebra is stupid": Did you ever notice that nobody asks why he "has" to take English Lit or phys-ed? But math and science are much more crucial to the basis of a modern technological society than are Moby Dick or the rules to dodge-ball. So why do we only hear complaints about math and science? Perhaps because they're hard...? Because they require work and discipline...? Because they aren't always "easy"...?
Modern educationist philosophy in America seems to say that education has to be "fun" and "entertaining" to be justifiable. Today's students often absorb the ethic that, unless a thing is easy, they shouldn't have to bother. But most worthwhile things in life are going to require some effort. If you want that great job, that interesting career, that open-ended future, you're almost certainly going to need some mathematical skills. And algebra is the basis, the foundation, the tool-box, for those skills.
"I'm only taking this class because the university makes me!": Let's be brutally honest here. The university didn't put a gun to your head and make you enroll. You decided you wanted their degree. You wanted their piece of paper.
Why? Probably so you could (eventually) get a better job. In order to get that job, you need at least some subset of the skills which are taught in algebra. You might be right that you'll never factor another quadratic in your entire life. But you want the university's piece of paper, so you're going to have to jump through the hoops required to get it. The algebra class is one of those hoops. If you don't want to jump through the hoop, that's fine; but you won't get the piece of paper. It's your choice.
"But I won't need this stuff for my job": A big difference between a student with an education and a worker with some training is the expectation that the student will have a deeper level of understanding, a broader base of knowledge, and a greater ability to build connections.
Will you, to a certainty, need everything taught in algebra? No. Does this mean that you should drop out of school now, get a job, and get only the training which is specific to your position?
"I can't drop out!", you reply, "I can't get that job unless I have a college degree." Ah. So, to get the job you want, you need to demonstrate proficiency in basic job skills. To demonstrate that proficiency, you need a degree. To get the degree, you need algebra. In other words, you do need this stuff for your job.
"Then I really will need algebra for 'real life'?": Maybe. Maybe not.
Consider the frequency with which "non-traditional" returning students have to take remedial math classes. The fact that they are taking algebra now, all these years past high school, strongly suggests that they haven't used algebra much in the years since they graduated. They got this far in life without algebra. But does that mean you shouldn't take algebra now?
The very fact that middle-aged folks are going back to college tells you that they need more than only what they'd previously been using in "real life". To move on, to move up, they need an education -- they need algebra. Take the hint.Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
"But why, exactly, do I have to take this stuff?": I have no idea. I don't know what degree you're pursuing; what your plans, hopes, or dreams are; or what your future might hold.
But consider: You didn't learn your alphabet all those years ago because you knew you'd be reading Moby Dick this semester. In the same way, you don't take algebra now because you know that you'll be factoring quadratics in ten years. You should take math and science courses now for much the same reason you learned your letters back then: to lay the foundation for bigger and better things to come, and to open up new opportunities for future pleasures and successes.
Nobody can say with assurance what skills will be needed twenty years from now. But what intelligent person would want to cut himself off from future opportunities and growth by refusing to expose himself to at least some of the knowledge which will be foundational for whatever is yet to come?
Even in the short term, you'll need some of the skills from algebra. If you're going to work with formulas in spreadsheets, you will need to be comfortable with variables and formulas. That's algebra. If you're going to be in meetings involving reports with tables, charts, and graphs, you'll need to be able to interpret these intelligently if you hope to hold your own in the discussions. That's algebra.
"Will algebra even be 'relevant' in the future?": While jobs and their specific skill-sets may change over time, mathematics won't. Twenty years from now, two plus two will still be four, and quadratics will still be either factorable or prime. Whatever job you get will provide the job-specific training you need, but to get that job in the first place, you're going to need some background knowledge and skills. And to be able to keep up with progress, to keep on top of new skill-sets, to move up the ladder, to jump across into a new and better career field, you will need the flexibility of a broad foundation. That foundation includes mathematics.
The philosopher Santayana famously said that "[t]hose who cannot learn from history are doomed to repeat it". This doesn't mean that you'd better memorize all those names and dates, or else long-dead people will rise from the grave and repeat everything they did before. It means that you need to learn the patterns and lessons of history, learn the cautionary tales to be gleaned from the (historical) mistakes of others, or else you may find yourself ignorantly making the same mistakes that those other people did. It is the lessons and patterns that are important.
The lessons and patterns of mathematics are important, too. If all you take from algebra is a comfort with variables and formulas, an ability to interpret graphs and to think logically, and a willingness to use abstraction when you try to solve problems, then you have gained some incredibly useful life skills, skills that will open doors, give you options, and allow you to make your own informed choices.
The specific algorithms you might study are not as important as the general patterns, techniques, and lessons that you can learn. Don't short-change your future by opting out now.
5/13/09
5/12/09
The point is that when we look at something like math and we don't see what we remember from school, we wonder about its validity. Did it really get "watered down" like all the news reports have been saying. Are modern kids being asked to do anything like the rigorous curriculum we used to work through? One precept of modern math instruction is that conceptual understanding not only supports math methods, but also should be the precursor much in the way understanding a cat from a holistic standpoint is the important first step before understanding it from a genetic perspective. Modern mathematics is always looking for the next conceptual model to study, so we are emulating that by focussing so much on modeling concepts like subtraction, fractions or quadratic equations. If we look at this from another point of view, human intellectual development follows a similar pathway from young child to adult.
5/8/09
5/6/09
5/3/09
Words that up-lift or shoot down
Words can be very uplifting, or they can tear you down. It was a tough week or two, and I was just not myself. One student stopped by to say hello and check on me, it was that simple. At that moment everything changed for me just a little bit, just enough.
Words carry a lot of power and we need to be careful, whether we are using email, or face-to-face.
We all need to:
- Think before we speak. If we think it may cause hurt, it probably will. Sleep on it and try to reword what we need to say.
- Never shoot from the hip (or lips).
- Try to put yourself in their shoes.
- Remember that the written word can be misunderstood, say it in person.
- Most importantly find a kind word to say whenever possible.
4/29/09
4/25/09
Margaret Wertheim: The beautiful math of coral (and crochet)
Best quote from this video: not a lot of mathematicians spend much time observing sea slugs: so they could never model hyperbolic space before a crocheter discovered it.
8th Grade Algebra?
While I do believe that 8th grade algebra is a good thing, it requires us teachers to ramp up the quality and rigor of arithmetic instruction in the previous grades:
Assumption Four: The United States should require all students to take algebra in the 8th grade and higher-order math in high school in order to increase the number of scientists and engineers in this country and thus make us more competitive in the global economy.
This assumption has become almost an obsession in policymaking arenas today. Requiring every student to study higher-order math is a waste of resources and cruel and unusual punishment for legions of students. It diverts attention away from the real problem: our failure to help kids become proficient readers and master basic arithmetic.
The United States must indeed produce more scientists and engineers to compete in a global economy. But it is fallacious to assume that we can accomplish that by requiring every student to take algebra in the 8th grade and higher-order math through high school. It is like believing that by requiring high school students to take a few courses in painting, we will make them all artists.
Most young people who go into science and engineering are well on their way by the time they start high school, because they become hooked on science or math in the early grades and do well in mathematics in elementary and middle school. Some will go on to become scientists and engineers; others will not. To expect otherwise is unreasonable.
If the nation wants more scientists and engineers, then educators need to find ways to awaken and nourish a passion for those subjects well before high school, and then offer students every opportunity to pursue their interest as far as they wish.
Assumption Four: The United States should require all students to take algebra in the 8th grade and higher-order math in high school in order to increase the number of scientists and engineers in this country and thus make us more competitive in the global economy.
This assumption has become almost an obsession in policymaking arenas today. Requiring every student to study higher-order math is a waste of resources and cruel and unusual punishment for legions of students. It diverts attention away from the real problem: our failure to help kids become proficient readers and master basic arithmetic.
The United States must indeed produce more scientists and engineers to compete in a global economy. But it is fallacious to assume that we can accomplish that by requiring every student to take algebra in the 8th grade and higher-order math through high school. It is like believing that by requiring high school students to take a few courses in painting, we will make them all artists.
Most young people who go into science and engineering are well on their way by the time they start high school, because they become hooked on science or math in the early grades and do well in mathematics in elementary and middle school. Some will go on to become scientists and engineers; others will not. To expect otherwise is unreasonable.
If the nation wants more scientists and engineers, then educators need to find ways to awaken and nourish a passion for those subjects well before high school, and then offer students every opportunity to pursue their interest as far as they wish.
Presenting at the National Council of Teachers of Mathematics Council 2009
Washington DC is a gorgeous city!!!
It was an interesting experience presenting original material in a national conference. I was prepared for 25-30 participants and had developed the workshop with that in mind. The conference organizers placed me early afternoon on Thursday in a room with a capacity of almost 400! I had to spend most of Thursday morning trying to re-organize my presentation and traipsed around town to make more copies. In the end I had around 100 participants, so I was more than prepared with the copies but still not quite with the format of the workshop.
Overall, I would say that the workshop went fine, though not as I had planned. From what I could tell, many participants were positive and active with the materials and genuinely interested in doing Problems of the Week with their students. I have received emails of appreciation and further inquiry from teachers all over the country.
As a learning experience, you can’t cross a chasm with two medium steps, but rather with one large leap, so I have no regrets, but many points I will be reflecting on. The biggest take away is that our society has moved pretty far away from paper: the best workshops were Powerpoint visuals with email addresses and links to materials from websites. That is where I want to be and learned I should be.
As a networking experience, nothing beat talking to educators around the country to figure out our relative position in the bigger scheme of things. 8th grade algebra is not as widespread as I would have hoped. In other places, 7th grade is being taught algebra first, then repeated in 8th grade in order to pass the state tests. I found that horrifying. If there was one message to be taken from all the workshop presenters, it is that this should not be the case. But political realities are as they are.
I met one of the lead publishers for our Algebra program and had a conversation about their other work with 6th and 7th grades. I also talked with a staff developer for our current 6th and 7th grade program who gave me some very good tips about their website help for parents. Most importantly, I met the Problem of the Week team from Mathforum.org out of Drexel University, who I have been talking and exchanging emails with for years. It was nice to put names to faces in this digital world.
I spoke briefly with the publisher of the TERC curriculum that the school adopted years ago. There is a newer version of the curriculum that I found engrossing and family friendly. I think we should consider using these materials in the 4th and 5th grades in the years to come and make an effort to properly learn how to use them to the best effect.
The single most exciting workshop I attended was called Using Web 2.0 to Teach Math 2.0. It was organized by a professor out of Washington State and three of her grad students. A key message is that students benefit when teachers learn how to use the communication capabilities of today’s Web. As one grad student said: “Email is so ‘last century’”. Each grad student showcased a particular project they used with their students that promoted collaboration and communication in the format that young people relate to, such as Facebook, Twitter, blogs, and many more and cooler things. This workshop really spoke to me as I continue my blogging efforts and have incorporated Twitter as a principal source of professional development.
One interesting quote from an article they pointed to:
“We are challenged to think about how to best prepare our kids for the ‘hypertransparent and hyperconnected world’ in which they are going to work and play… in this environment, ‘how’ you do something is more important even than ‘what’ you do. If you’re not doing it skillfully, ethically, and transparently, you’ve be ceding success to those that do. It is not how good your resume is, but rather, how well you come out in a Google search that counts.”
It was an interesting experience presenting original material in a national conference. I was prepared for 25-30 participants and had developed the workshop with that in mind. The conference organizers placed me early afternoon on Thursday in a room with a capacity of almost 400! I had to spend most of Thursday morning trying to re-organize my presentation and traipsed around town to make more copies. In the end I had around 100 participants, so I was more than prepared with the copies but still not quite with the format of the workshop.
Overall, I would say that the workshop went fine, though not as I had planned. From what I could tell, many participants were positive and active with the materials and genuinely interested in doing Problems of the Week with their students. I have received emails of appreciation and further inquiry from teachers all over the country.
As a learning experience, you can’t cross a chasm with two medium steps, but rather with one large leap, so I have no regrets, but many points I will be reflecting on. The biggest take away is that our society has moved pretty far away from paper: the best workshops were Powerpoint visuals with email addresses and links to materials from websites. That is where I want to be and learned I should be.
As a networking experience, nothing beat talking to educators around the country to figure out our relative position in the bigger scheme of things. 8th grade algebra is not as widespread as I would have hoped. In other places, 7th grade is being taught algebra first, then repeated in 8th grade in order to pass the state tests. I found that horrifying. If there was one message to be taken from all the workshop presenters, it is that this should not be the case. But political realities are as they are.
I met one of the lead publishers for our Algebra program and had a conversation about their other work with 6th and 7th grades. I also talked with a staff developer for our current 6th and 7th grade program who gave me some very good tips about their website help for parents. Most importantly, I met the Problem of the Week team from Mathforum.org out of Drexel University, who I have been talking and exchanging emails with for years. It was nice to put names to faces in this digital world.
I spoke briefly with the publisher of the TERC curriculum that the school adopted years ago. There is a newer version of the curriculum that I found engrossing and family friendly. I think we should consider using these materials in the 4th and 5th grades in the years to come and make an effort to properly learn how to use them to the best effect.
The single most exciting workshop I attended was called Using Web 2.0 to Teach Math 2.0. It was organized by a professor out of Washington State and three of her grad students. A key message is that students benefit when teachers learn how to use the communication capabilities of today’s Web. As one grad student said: “Email is so ‘last century’”. Each grad student showcased a particular project they used with their students that promoted collaboration and communication in the format that young people relate to, such as Facebook, Twitter, blogs, and many more and cooler things. This workshop really spoke to me as I continue my blogging efforts and have incorporated Twitter as a principal source of professional development.
One interesting quote from an article they pointed to:
“We are challenged to think about how to best prepare our kids for the ‘hypertransparent and hyperconnected world’ in which they are going to work and play… in this environment, ‘how’ you do something is more important even than ‘what’ you do. If you’re not doing it skillfully, ethically, and transparently, you’ve be ceding success to those that do. It is not how good your resume is, but rather, how well you come out in a Google search that counts.”
4/24/09
4/21/09
3/29/09
| The Colbert Report | Mon - Thurs 11:30pm / 10:30c | |||
| Math Is Hard | ||||
| ||||
3/17/09
3/15/09
This is a huge bummer and unfair to CA teachers and students!
SAN JOSE, Calif. (AP) — Public school employees throughout California are being warned of extensive job cuts as local officials face a deadline for issuing layoff notices to educators.
The State Department of Education estimates that preliminary notices will be given to 26,500 teachers by the Sunday cutoff. An additional 15,000 bus drivers, janitors, secretaries and administrators are also expected to receive the written warnings.
Because of the bleak economic outlook, the state has ordered local school districts to absorb more than $8 billion in cuts over the next year. Many, if not most, of the early layoff notices could be withdrawn by June, officials said.
The State Department of Education estimates that preliminary notices will be given to 26,500 teachers by the Sunday cutoff. An additional 15,000 bus drivers, janitors, secretaries and administrators are also expected to receive the written warnings.
Because of the bleak economic outlook, the state has ordered local school districts to absorb more than $8 billion in cuts over the next year. Many, if not most, of the early layoff notices could be withdrawn by June, officials said.
This is a huge bummer and unfair to CA teachers and students!
SAN JOSE, Calif. (AP) — Public school employees throughout California are being warned of extensive job cuts as local officials face a deadline for issuing layoff notices to educators.
The State Department of Education estimates that preliminary notices will be given to 26,500 teachers by the Sunday cutoff. An additional 15,000 bus drivers, janitors, secretaries and administrators are also expected to receive the written warnings.
Because of the bleak economic outlook, the state has ordered local school districts to absorb more than $8 billion in cuts over the next year. Many, if not most, of the early layoff notices could be withdrawn by June, officials said.
The State Department of Education estimates that preliminary notices will be given to 26,500 teachers by the Sunday cutoff. An additional 15,000 bus drivers, janitors, secretaries and administrators are also expected to receive the written warnings.
Because of the bleak economic outlook, the state has ordered local school districts to absorb more than $8 billion in cuts over the next year. Many, if not most, of the early layoff notices could be withdrawn by June, officials said.
Trillion Dollar Deficits
To the average person, a number that big probably doesn't mean much. At some point long before the hundred-billion-dollar mark, large numbers simply become figures on the page, well beyond human scale and intuitive understanding. Is a $700 billion financial-industry bailout a lot? Is a $775 billion economic-stimulus package enough?
Unfortunately, our puny human brains aren't particularly up to the task. Go back thousands of years and think about the simpler times of human existence. "We had a few friends; we had to be scared of a few animals. A trillion didn't come up very often," says Temple University mathematician John Allen. "There is a sense that when numbers are too big or too small, the brain just shuts off."
The genius of our numbering system is that we can signify massive quantities in short spaces. One billion takes no longer to write than one million does.
But that similarity trips us up when it comes time to imagine how those figures translate to the real world, where three more zeros make all the difference. "My favorite way to think of it is in terms of seconds," says David Schwartz, a children's book author whose How Much Is a Million? tries to wrap young minds around the concept. "One million seconds comes out to be about 11½ days. A billion seconds is 32 years. And a trillion seconds is 32,000 years. I like to say that I have a pretty good idea what I'll be doing a million seconds from now, no idea what I'll be doing a billion seconds from now, and an excellent idea of what I'll be doing a trillion seconds from now."
A common strategy for beginning to understand big numbers is to devise visual representations. One time, sitting at a baseball game in Philadelphia, Paulos started counting seats along the first-base line. Multiplying the number of seats in a row by the number of rows, Paulos came up with a section of the stadium that he figured contained about 10,000 seats — an image he can now think back to whenever a person starts talking about tens of thousands of a particular thing. When numbers get too large, though, that method breaks down. A stack of one trillion $1 bills would reach more than a quarter of the way to the moon — replacing one incomprehensible thought with another doesn't do much good.
We next move on to more formal manipulations. When trying to comprehend a trillion-dollar deficit, you might calculate how much money that represents per person in the U.S. One trillion dollars divided by 300 million Americans comes out to $3,333. Then you search for a useful comparison. A convenient — though perhaps unsettling — comparison is to the amount of credit-card debt carried by the average person in this country. That figure is $3,245. "So a good way of thinking about government debt financing is that it's similar to what the average person is doing," says Camerer.
Unfortunately, our puny human brains aren't particularly up to the task. Go back thousands of years and think about the simpler times of human existence. "We had a few friends; we had to be scared of a few animals. A trillion didn't come up very often," says Temple University mathematician John Allen. "There is a sense that when numbers are too big or too small, the brain just shuts off."
The genius of our numbering system is that we can signify massive quantities in short spaces. One billion takes no longer to write than one million does.
But that similarity trips us up when it comes time to imagine how those figures translate to the real world, where three more zeros make all the difference. "My favorite way to think of it is in terms of seconds," says David Schwartz, a children's book author whose How Much Is a Million? tries to wrap young minds around the concept. "One million seconds comes out to be about 11½ days. A billion seconds is 32 years. And a trillion seconds is 32,000 years. I like to say that I have a pretty good idea what I'll be doing a million seconds from now, no idea what I'll be doing a billion seconds from now, and an excellent idea of what I'll be doing a trillion seconds from now."
A common strategy for beginning to understand big numbers is to devise visual representations. One time, sitting at a baseball game in Philadelphia, Paulos started counting seats along the first-base line. Multiplying the number of seats in a row by the number of rows, Paulos came up with a section of the stadium that he figured contained about 10,000 seats — an image he can now think back to whenever a person starts talking about tens of thousands of a particular thing. When numbers get too large, though, that method breaks down. A stack of one trillion $1 bills would reach more than a quarter of the way to the moon — replacing one incomprehensible thought with another doesn't do much good.
We next move on to more formal manipulations. When trying to comprehend a trillion-dollar deficit, you might calculate how much money that represents per person in the U.S. One trillion dollars divided by 300 million Americans comes out to $3,333. Then you search for a useful comparison. A convenient — though perhaps unsettling — comparison is to the amount of credit-card debt carried by the average person in this country. That figure is $3,245. "So a good way of thinking about government debt financing is that it's similar to what the average person is doing," says Camerer.
4/7/08
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