Happy Thanksgiving!
My algebra class is winding down. We only have two more meetings. One for review. One for a final.
I'm doing fine in the course, but many of my classmates aren't. Many of them may not get the "C" required to move on the the next course, College Algebra, that is required for them to be accepted into many majors, including business.
One kid, a scholarship golfer, stopped coming. I hope he didn't drop the class. I've thought about him and learning math during the semester. I've learned, during the semester, that it is about about 100 times easier to go through the steps to factor an equation than it is to hit a 300 yard drive down middle of the fare way and probably 1,000 times easier to learn how to plug numbers into the quadratic equation than it is to sink a 12 foot putt for birdie.
But he can't seem to factor or solve for x using the quadratic equation.
Out at his country club during practice, he must drive hundreds of balls a day to get it right. He must practice his putting for hours. If he didn't, he wouldn't be capable of shooting two under, like he does.
I've learned this semester that math, at this level, is pretty much mechanical -- following a recipe. First you do this, then you do this, then you do that, and pretty soon you have an answer and you're done. I've found that to be both good news and bad news. Good news because, if I'm willing to put in the time and follow the recipe over and over I can do the stuff. Bad news because it really, so far, doesn't require any imagination, insight, or inspiration.
At the same time, I tell myself, when you are learning a new language in a classroom (rather than in a bar or in bed) learning conjugation or tenses doesn't require a lot of imagination either.
A devine intervention (real time)
It was almost as if god read the paragraph I just wrote and, through the internet, said, "Bullshit."
So, seconds ago, as I was distracting myself from writing, I clicked over to check my email. Waiting for me was a message, not there five minutes earlier (last time I checked) from Paul Niquette.
Paul is someone I've been intending to introduce you to since I began this project. He is one of the smartest and most inspirational people I'll ever meet. You can read about him on his web site and I'll tell you more during a later post.
He and I, over the years have talked about many things. Bicycling. Books. Oil. Travel. Trains. Words, and Language. But we've never talked about math. And that is something he thinks about a lot. As soon as I was done with this semester's classes, I was going to call him and ask if I could tape a conversation about math and numbers. I hope I still get to.
Minutes ago, he sent me and my family some good Thanksgiving wishes along with what looked like a news release about what he had been up to. It seems that he and his firm have been involved in passing "Measure B," a ballot measure that would increase the sales tax to support the "Silicon Valley Rapid Transit Corridor through 2036." It needed to pass by a 2/3 majority (and did).
He included a teaser with his news -- a word problem. Here it is:
"On November 18th, 612,ooo votes had been counted, and barely 66.67% were in favor of Measure B. However, there were still 9,800 votes left to count. According to the last report on November 25th, Measure B was declared to have passed with 66.78% of the votes. What is the largest possible number of uncounted ballots?"
I decided to tackle the problem. So, with only a little help from Gail, I figured out that there could be as many as 2582 ballots left uncounted for the measure to have passed by 66.78%. And I couldn't believe I did the problem with just a little imagination, some insight, and a touch of inspiration.
Seth is coming over in a few minutes. I can't wait to ask him if I got the right answer.
Thursday, November 27, 2008
Sunday, November 23, 2008
Math Thoughts from Jake Marcus
Jake Marcus landed on this blog because his aunt Margy Rochlin told him about it.
Margy is my first cousin and a full time writer. She and I were talking about my algebra class and my blog and writing. Jake graduated from Yale in spring 2008 with a degree in math so it was natural that Margy mention my experiment.
Jake thinks about the arts and philosophy and other countries and cultures, and he thinks a lot about math.
So Jake visited here and wrote a long letter to me that I want to share parts of with you. There was a lot there, so I'll try to share only some of the highlights in bite size chunks.
Jake's words are in purple.
---------------------------------------------------------------
Math is the product of making ideas as precise as possible.
If you take any argument and try to clarify its assumptions, define all of its terms, strengthen each step of the argument so that any one step indisputably leads to the next, then you will find yourself doing math.
What really got my attention was that Jake was thinking about "ideas and precision" rather than "counting and precision." I'm sure I can't yet appreciate what he is thinking, but I want to and I think that those couple of sentences alone might sum up a good part of my attraction to math as a subject.
One other thing Jake wrote summed up something else I've been thinking about:
A lot of the creativity in math is picking the right definition for what you are interested in exploring. Once you decide on that definition, then there is one right answer for what that definition implies. Of course, there are many ways to pick that definition in the first place, some much more reasonable than others, but no one right definition
Jake went on to make a point about precision and the importance of definitions using "volume" as an example. (Jake's writing is fairly clear, but it is still easy to get lost. I did.)
The field tends to focus on simple and idealized concepts like triangles, distance or whole numbers, because it takes so much effort to make even these deceptively simple ideas precise.
Take the concept of volume.
We all have a good idea of what volume means and sure we toss the word around idly in conversation. But what if we want to make volume precise?
We might start by looking it up in the dictionary. Merriam-Webster defines volume as the amount of space occupied by a three-dimensional object. This definition suffices for daily living, but not if we’re sticklers about it. We need a definition that will allow us to make unassailable arguments about volume and its properties.
If we use Merriam-Webster’s definition, someone will inevitably come along and question what we mean by “object” or “space” and use that ambiguity to poke holes in the arguments we try to make about, say, the volume of spheres or donuts or any manner of exotic and absurd three-dimensional shapes.
Let’s use some mathematics to think about volume.
First, decide on some reference point in your house, for example the rightmost atom on your living room sofa and assign it the numbers 0, 0, 0. Now, assign to every point in the entire universe three numbers.
The distance from your reference point walking in the horizontal direction, the distance from your reference point walking in the vertical direction and the distance from your reference point walking upwards towards the sky (imagining that you can walk that way).
Some of these numbers will be very small. You might assign the atom just to the right of your reference point, the numbers .00000001, 0, 0 and the atom just to the left of your reference point, the numbers 0, .000000001, 0 and the atom just above your reference point, 0, 0, .000000001, but you might assign the numbers 1000000, 2000000, 1000000 to some atom in your bedroom and assign some really very big numbers to the points off in far corners of the milky way.
Now we have an x-axis, a y-axis and a z-axis for the universe and can start thinking about collections of those points, or x, y and z coordinates.
The collection of all the points a distance of one or less away from the rightmost atom on your living room sofa is a sphere with a radius of one centered on that rightmost atom.
You can think about any collection of points you want. You can cut out cubes, pyramids, cylinders and cones from the earth as if you were cutting shapes out of fabric.
The question then becomes, how do we assign to each collection of points a number, called volume?
Whatever process we decide on for assigning a number (volume) to different collections of points will in effect define the concept of volume for us.
Suppose you collect some points together and they make a cube of length 2, width 3 and height 4. Then we want our process to assign the number 2x3x4 or 24 to this particular cube and to assign the product of the length, width and height to cubes in general. We can probably agree on a few other properties that our volume-assigning process must have.
There are four properties in particular that seem reasonable for our process to preserve no matter the collection of points:
1) Volume should never be negative, it should be 0, infinity or some number in between.
2) The volume of two distinct objects put together should be the sum of the volume of those two objects separated.
3) A sphere with a radius bigger than 0 should have a volume somewhere between 0 and infinity.
4) If you can make one object exactly the same as the other by rotating it or moving it around, then the volume of those two objects should be the same.
It seems like whatever process we decide on for assigning a volume to different collections of points should observe these four rules.
But here’s the punch line: no process exists that observes those four rules!
No one will ever discover one either. Mathematicians have proved that no such process can exist.
As soon as you imagine a way of assigning volume to objects that always observes one of those rules, it contradicts another. If you assume a process exists that sticks to rules 1-3, then the Banach-Tarski Paradox shows that this process does not observe rule 4, that is a ball can be taken apart into five pieces, each of these pieces can be rotated and moved around, put back together and the ball will be bigger than it was. Granted these 5 pieces have to be really weird shapes that couldn’t be practically constructed (in the real world, we know how to cut a ball in half, but not how to cut a ball in 1/Ö2).
There is no one way to define volume and the task of making whatever definition you decide on precise is a subtle one.
Jake's last point he wanted to make to me was motivating, if not daunting.
Calculus gives some very good definitions for concepts that seem ethereal and unclear. It is not any less precise than other forms of mathematics; rather its strength lies in making precise concepts like infinity, continuous, smooth and infinitesimal.
Jake gave an example from Zeno’s paradoxes. If you're interested in reading about the paradoxes, here's a link.
I'll finish this entry by letting Jake finish:
In mathematics beyond a certain point, how you approach a problem does, I think, matter the most.
Some very famous mathematicians made their discoveries by making connections between fields thought to be completely unrelated.
Evariste Galois saw the connection between mathematical objects called fields, groups and polynomials and with this insight solved geometrical problems that had gone unsolved for thousands of years.
Since the Greeks, mathematicians had wondered with it was possible to trisect an angle (split an angle up into three equal angles) using only a compass and a ruler. With Galois theory, you can prove that it’s impossible.
I remember taking Galois theory and being amazed that the professor showed us how to solve three problems in one lecture that had taken humanity thousands of years to figure out.
Jake, thanks for your thoughts!
Margy is my first cousin and a full time writer. She and I were talking about my algebra class and my blog and writing. Jake graduated from Yale in spring 2008 with a degree in math so it was natural that Margy mention my experiment.
Jake thinks about the arts and philosophy and other countries and cultures, and he thinks a lot about math.
So Jake visited here and wrote a long letter to me that I want to share parts of with you. There was a lot there, so I'll try to share only some of the highlights in bite size chunks.
Jake's words are in purple.
---------------------------------------------------------------
Math is the product of making ideas as precise as possible.
If you take any argument and try to clarify its assumptions, define all of its terms, strengthen each step of the argument so that any one step indisputably leads to the next, then you will find yourself doing math.
What really got my attention was that Jake was thinking about "ideas and precision" rather than "counting and precision." I'm sure I can't yet appreciate what he is thinking, but I want to and I think that those couple of sentences alone might sum up a good part of my attraction to math as a subject.
One other thing Jake wrote summed up something else I've been thinking about:
A lot of the creativity in math is picking the right definition for what you are interested in exploring. Once you decide on that definition, then there is one right answer for what that definition implies. Of course, there are many ways to pick that definition in the first place, some much more reasonable than others, but no one right definition
Jake went on to make a point about precision and the importance of definitions using "volume" as an example. (Jake's writing is fairly clear, but it is still easy to get lost. I did.)
The field tends to focus on simple and idealized concepts like triangles, distance or whole numbers, because it takes so much effort to make even these deceptively simple ideas precise.
Take the concept of volume.
We all have a good idea of what volume means and sure we toss the word around idly in conversation. But what if we want to make volume precise?
We might start by looking it up in the dictionary. Merriam-Webster defines volume as the amount of space occupied by a three-dimensional object. This definition suffices for daily living, but not if we’re sticklers about it. We need a definition that will allow us to make unassailable arguments about volume and its properties.
If we use Merriam-Webster’s definition, someone will inevitably come along and question what we mean by “object” or “space” and use that ambiguity to poke holes in the arguments we try to make about, say, the volume of spheres or donuts or any manner of exotic and absurd three-dimensional shapes.
Let’s use some mathematics to think about volume.
First, decide on some reference point in your house, for example the rightmost atom on your living room sofa and assign it the numbers 0, 0, 0. Now, assign to every point in the entire universe three numbers.
The distance from your reference point walking in the horizontal direction, the distance from your reference point walking in the vertical direction and the distance from your reference point walking upwards towards the sky (imagining that you can walk that way).
Some of these numbers will be very small. You might assign the atom just to the right of your reference point, the numbers .00000001, 0, 0 and the atom just to the left of your reference point, the numbers 0, .000000001, 0 and the atom just above your reference point, 0, 0, .000000001, but you might assign the numbers 1000000, 2000000, 1000000 to some atom in your bedroom and assign some really very big numbers to the points off in far corners of the milky way.
Now we have an x-axis, a y-axis and a z-axis for the universe and can start thinking about collections of those points, or x, y and z coordinates.
The collection of all the points a distance of one or less away from the rightmost atom on your living room sofa is a sphere with a radius of one centered on that rightmost atom.
You can think about any collection of points you want. You can cut out cubes, pyramids, cylinders and cones from the earth as if you were cutting shapes out of fabric.
The question then becomes, how do we assign to each collection of points a number, called volume?
Whatever process we decide on for assigning a number (volume) to different collections of points will in effect define the concept of volume for us.
Suppose you collect some points together and they make a cube of length 2, width 3 and height 4. Then we want our process to assign the number 2x3x4 or 24 to this particular cube and to assign the product of the length, width and height to cubes in general. We can probably agree on a few other properties that our volume-assigning process must have.
There are four properties in particular that seem reasonable for our process to preserve no matter the collection of points:
1) Volume should never be negative, it should be 0, infinity or some number in between.
2) The volume of two distinct objects put together should be the sum of the volume of those two objects separated.
3) A sphere with a radius bigger than 0 should have a volume somewhere between 0 and infinity.
4) If you can make one object exactly the same as the other by rotating it or moving it around, then the volume of those two objects should be the same.
It seems like whatever process we decide on for assigning a volume to different collections of points should observe these four rules.
But here’s the punch line: no process exists that observes those four rules!
No one will ever discover one either. Mathematicians have proved that no such process can exist.
As soon as you imagine a way of assigning volume to objects that always observes one of those rules, it contradicts another. If you assume a process exists that sticks to rules 1-3, then the Banach-Tarski Paradox shows that this process does not observe rule 4, that is a ball can be taken apart into five pieces, each of these pieces can be rotated and moved around, put back together and the ball will be bigger than it was. Granted these 5 pieces have to be really weird shapes that couldn’t be practically constructed (in the real world, we know how to cut a ball in half, but not how to cut a ball in 1/Ö2).
There is no one way to define volume and the task of making whatever definition you decide on precise is a subtle one.
Jake's last point he wanted to make to me was motivating, if not daunting.
Calculus gives some very good definitions for concepts that seem ethereal and unclear. It is not any less precise than other forms of mathematics; rather its strength lies in making precise concepts like infinity, continuous, smooth and infinitesimal.
Jake gave an example from Zeno’s paradoxes. If you're interested in reading about the paradoxes, here's a link.
I'll finish this entry by letting Jake finish:
In mathematics beyond a certain point, how you approach a problem does, I think, matter the most.
Some very famous mathematicians made their discoveries by making connections between fields thought to be completely unrelated.
Evariste Galois saw the connection between mathematical objects called fields, groups and polynomials and with this insight solved geometrical problems that had gone unsolved for thousands of years.
Since the Greeks, mathematicians had wondered with it was possible to trisect an angle (split an angle up into three equal angles) using only a compass and a ruler. With Galois theory, you can prove that it’s impossible.
I remember taking Galois theory and being amazed that the professor showed us how to solve three problems in one lecture that had taken humanity thousands of years to figure out.
Jake, thanks for your thoughts!
Sunday, November 16, 2008
Ethics and Math?
A journalism department colleague sent me email expressing concerns about something I am allowing or suggesting members of my class do. The substance of our discussion and disagreement has nothing to do with math, but it did get me thinking.
Is there an ethics of math?
Is there a philosophy of math?
Do any of the questions that those of who live in a qualitative or aesthetic world ask a hundred times a day have any place in the world of math?
I hear mathematicians talk about "beauty" but what are they really thinking about. Same with elegance.
There are, of course right and wrong answers to problems or equations. But do the notions of "right" and "wrong" in a moral sense have any place at all when examining the world through a mathematical lens?
Do pure mathematicians find religious meaning in some numbers?
I'd love to get Prof. Saari on the phone and ask him.
I'd love to get back in touch with Paul Niquette. He thinks about these things.
I guess, now, so do I.
Is there an ethics of math?
Is there a philosophy of math?
Do any of the questions that those of who live in a qualitative or aesthetic world ask a hundred times a day have any place in the world of math?
I hear mathematicians talk about "beauty" but what are they really thinking about. Same with elegance.
There are, of course right and wrong answers to problems or equations. But do the notions of "right" and "wrong" in a moral sense have any place at all when examining the world through a mathematical lens?
Do pure mathematicians find religious meaning in some numbers?
I'd love to get Prof. Saari on the phone and ask him.
I'd love to get back in touch with Paul Niquette. He thinks about these things.
I guess, now, so do I.
Tuesday, November 11, 2008
Understanding Change
I went in for my annual physical this morning. Dr. James Reifschneider asked me what I was up to. Among other things, I told him about taking algebra.
He became more interested than usual.
Then he gave me a lecture about how, in his view, you simply cannot begin to grasp change or the nature of change unless you could understand calculus. He said whether it's about biology, astronomy, physics, or whatever, you need calculus to have a clue about what's going on.
He went on to say that earlier in the year he had purchased a book called Calculus for Dummies
and was working his way through it.
---------------------------------------------
Meanwhile, back in class, we had another test and I think I did pretty well. Afterward Prof. John said that I might be in the lead for "most improved player." I got a good laugh out of that because Seth says that is probably the least coveted award -- in the sports world anyway.
He became more interested than usual.
Then he gave me a lecture about how, in his view, you simply cannot begin to grasp change or the nature of change unless you could understand calculus. He said whether it's about biology, astronomy, physics, or whatever, you need calculus to have a clue about what's going on.
He went on to say that earlier in the year he had purchased a book called Calculus for Dummies
and was working his way through it.---------------------------------------------
Meanwhile, back in class, we had another test and I think I did pretty well. Afterward Prof. John said that I might be in the lead for "most improved player." I got a good laugh out of that because Seth says that is probably the least coveted award -- in the sports world anyway.
Sunday, November 9, 2008
Pushing on
(Slow) Progress
I've been thinking and doing more about learning math and less about posting during the past week. Even though I did well on the last test, this week, I felt like I was falling behind.
I spent way too much time at the computer doing practice exercises.
As a result, I now get it about "rational expressions and functions" and "complex rational expressions" and "rational equations" and "division of polynomials."
We have a test on Monday covering all that plus factoring and inequalities.
I'm doing and understanding things that I didn't even know existed back at the end of August. I'm enjoying that, but just so that it doesn't go to my head, Seth reminds me that I am working on about the same stuff that his high school freshmen and sophomores are doing in his classes. However, I point out to him, they are the smart freshmen and sophomores. The ones that aren't in his classes are taking this same stuff with me as high school freshmen.
Just to turn up the pressure, everyone taking Pima College math 122 got an email telling us that if we don't get a "C" or better in this class, we won't be eligible to enroll in college algebra at the University of Arizona.
Perfectionism (not)
As the numbers or equations or functions get more and more intimidating, or at least scarier looking, I'm actually understanding what I need to so. I can set problems up and I know what has to go where, but still, too often, I mess up my arithmetic. I'm not sure I'm getting any better at that.
I've never been a perfectionist and never needed to be. Generally, I think it is a waste of time. You can get lots more done if everything you do doesn't have to be perfect. But in math, it seems to matter. I'm wondering (hoping?) if you get beyond a certain point, it stops mattering, and how you approach a problem matters more. That's one of the reasons I'm curious about calculus. I don't know anything at all about calculus, but in imagining, it doesn't seem that it is precise, that it can give one right answer. Otherwise, why would they need it? Or, I wonder whether most people who use calculus believe an outcome to be the one right answer based on faith rather than fact. I can't wait to find out in a couple of years.
A fun book
I've come across another book (recommended by Marcy, my UA math advisor). It's called The Joy of Mathematics: Discovering Mathematics All Around You. It contains 228 bite sized stories about things you might or might not have been curious about, all from a mathematical perspective.
Things I'm not doing
I've been thinking and doing more about learning math and less about posting during the past week. Even though I did well on the last test, this week, I felt like I was falling behind.
I spent way too much time at the computer doing practice exercises.
As a result, I now get it about "rational expressions and functions" and "complex rational expressions" and "rational equations" and "division of polynomials."
We have a test on Monday covering all that plus factoring and inequalities.
I'm doing and understanding things that I didn't even know existed back at the end of August. I'm enjoying that, but just so that it doesn't go to my head, Seth reminds me that I am working on about the same stuff that his high school freshmen and sophomores are doing in his classes. However, I point out to him, they are the smart freshmen and sophomores. The ones that aren't in his classes are taking this same stuff with me as high school freshmen.
Just to turn up the pressure, everyone taking Pima College math 122 got an email telling us that if we don't get a "C" or better in this class, we won't be eligible to enroll in college algebra at the University of Arizona.
Perfectionism (not)
As the numbers or equations or functions get more and more intimidating, or at least scarier looking, I'm actually understanding what I need to so. I can set problems up and I know what has to go where, but still, too often, I mess up my arithmetic. I'm not sure I'm getting any better at that.
I've never been a perfectionist and never needed to be. Generally, I think it is a waste of time. You can get lots more done if everything you do doesn't have to be perfect. But in math, it seems to matter. I'm wondering (hoping?) if you get beyond a certain point, it stops mattering, and how you approach a problem matters more. That's one of the reasons I'm curious about calculus. I don't know anything at all about calculus, but in imagining, it doesn't seem that it is precise, that it can give one right answer. Otherwise, why would they need it? Or, I wonder whether most people who use calculus believe an outcome to be the one right answer based on faith rather than fact. I can't wait to find out in a couple of years.
A fun book
I've come across another book (recommended by Marcy, my UA math advisor). It's called The Joy of Mathematics: Discovering Mathematics All Around You
. It contains 228 bite sized stories about things you might or might not have been curious about, all from a mathematical perspective.Things I'm not doing
I called my aunt, Harriet Rochlin, to wish her a happy 85th birthday. She is finishing a book about Jewish women. She has speaking commitments. She has an ongoing research agenda. She says she doesn't have a moment to just reflect and says he hopes that someday she will.
The more I get into this math adventure, the more I feel my list of things to do grow. Mostly little things, some bigger. Here's what I need to do, but haven't (just about this small part of my life):
1. Work on a book proposal about math for journalists.
2. Think about a book about a complete beginner learning about math.
3. Write to Prof. Saari to ask about all kinds of things, and just get a correspondence going.
4. Thank Dr. Jerry for having lunch with me and sharing his story. And ask him whether we could have a formal interview sometime so I can get details on tape and share them with you.
5. Email Prof. Velez to ask for a visit or lunch to talk about math
6. Email Prof. Alexander to ask whether he would be willing to share his story about how a submarine commander became a math professor.
7. Thank Marcy again for visiting with me and being such a good advisor.
8. Get accepted to the University of Arizona again so I can sign up for Math 112, college algebra, for spring semester.
9. Read The Joy of Mathematics
10. Read How to Lie with Statistics
11. Read Math Tools for Journalists
12. Read Numbers in the Newsroom
13. Learn more about my fancy HP calculator
14. And Post to this Blog more frequently
Then, there is tomorrow's test to prepare for.
Sunday, November 2, 2008
Mars and More - Dr. Jerry
Dr. Gerard Droege
Friday I enjoyed meeting and having a fascinating lunch with Dr. Gerard Droege, a fellow old guy taking math as an undergraduate.
Hopefully, I'll do a real interview with him sometime, so I'll give a full report then, but I just wanted to introduce him now. Some of the facts here might not be exactly correct because I didn't tape our conversation or even take notes.
Jerry looks to be in his early 50s with already grey hair but a young looking face. He still has young skin. He probably didn't spend much time in the sun as a young guy. He comes on gentle, reserved, but confident and happy enough. I liked him immediately when he came to my office before lunch.
Jerry spend most of his medical career as a small town OBGYN on the east coast. For years he was the only OB in his county and pretty much everyone who needed help, counted on him to be there. He said he never got a full night's sleep. At minimum, there would be phone calls, but often enough he had to head to the county hospital. The baby wouldn't wait.
He spent some time in San Diego working with the poor, but that got difficult once California passed a resolution forbidding Medicaid from paying for the delivery and births of children born to undocumented people.
During all this time Jerry kept up a fascination with math and astronomy.
Finally, after too many sleepless nights and too many hours spent just earning enough money to pay next year's insurance premiums, Jerry decided to go after a dream he was still forming.
But, he knew the dream had to do with math and astronomy. That's how he got to Arizona.
After 22 years in full time practice, Jerry quit medicine on Dec. 31. (If he worked even one day during the following year, the insurance would have cost him $100,000.) On the advice of a trusted scientist friend Jerry decided to apply to the University of Arizona as an undergraduate math major. He was accepted for the fall semester.
Jerry, who is not married and doesn't have children, took a little time to wrap things up and headed to Costa Rica. He spent six months on the beach reading math books and studying about astronomy.
He came to Tucson and registered for calculus. He had taken it in college, but figured he needed a refresher.
Just as he landed in Tucson, the Phoenix Mars Mission was getting under way. Jerry wanted to be part of it. With what I imagine was a perfect combination of chutzpah and naivety, Jerry went to mission headquarters and asked for a job.
Probably realizing there was not that great a need for a burned out OBGYN at mission control, the folks there turned him down.
But, as luck (and good thinking) would have it, Jerry came across a grant to fund students to work in the sciences and astronomy. He applied and won.
He went back and said he had secured his own funding and really wanted to be part of the Mars team. Would they hire him? This time, the answer was yes.
So, Jerry has had quite a ride. He became in integral part of the team with real responsibilities. He says, four or so years ago neither he nor any of his friends or colleagues could have imagined that he would be part of the group responsible for landing a satellite on Mars to search for life.
Nor could he have even begun to imagine that, on the surface of Mars, along with seven others is the name, Gerard Droege inscribed on a sheet of gold.
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