In my new grad school class called Technology and Learning, Gary Stager assigned us, his students, the task of composing a piece of music. The assignment asked:
Between now and next Monday, download Finale Notepad from http://www.finalenotepad.com and compose a piece of music. Share your reflections, questions and work-in-progress with your cadre members. Think about your own thinking and learning along the way. Make notes, keep a journal or blog if you wish. Feel free to use any resources at your disposal for inspiration or assistance. The more you share, the more you will learn. A few considerations: I realize there are better more powerful pieces of music composition available. However, Finale Notepad is 1) Free and 2) Cross-platform. Therefore, NO you can't use another piece of software (at least visible to the rest of us). Using a common tool provides a common experience and language for assisting/inspiring one another. Everyone has different experience levels and areas of expertise. This is what makes the learning adventures interesting.
I decided that rather than me creating the piece of music, it would be a great opportunity to connect music with literature. Gary gave me the suggestion of having the kids connect the piece of music to something they have written. Since we are reading William Shakespeare’s Macbeth in class, we had just completed two papers dealing with Macbeth and Lady Macbeth while watching three film versions (Royal Shakespeare Company -Ian McKellan, Dame Judy Dench, English Shakespeare Company- modern version, and Roman Polansky version) of the first act comparing what each director was trying to show through their interpretations.
My students were quite excited to begin the process of composing a piece of music to fit a character. In period 2, we started talking about Lady Macbeth and her personality. What would we want to show in the song? We decided that we would show her in three stages from act one. We would show her elation at seeing her husband return home, her decision to not let King Duncan leave the castle by killing him so that Macbeth would become king, and finally her anger at Macbeth with his ambivalence about killing the king. We had some students who are as musically inclined as I am and so they worked on the lyrics (a.k.a. quotes) to fit with the music as well as finding visuals to support. The other kids, worked on the song. Amazingly, they all worked so well together testing sounds, putting together measure after measure, playing the notes, time and again. They decided the instruments that would best describe Lady Macbeth (flute) and how her tone would change into a French horn through her change in personality. Then they picked the instruments that would best accompany the sounds of Lady Macbeth. A few different kids took turns running the computer with other kids shouting up their thoughts. It seemed like organized chaos. One thing I must add here is how much I learned by watching and participating with them. I learned all sorts of vocabulary word about music (crescendo, decrescendo, staccato) and how to semi-compose music (you really have to pay attention to the notes you select with each instrument). But mostly, what I am hoping period two took away from today, and what they learned, is that music is a part of literature. When talking with them about the song, they asked my opinion about a particular part. I said it needs to sound like murder. Tristan responded that murder is an A and C sharp. And then another student, John, responding that we need to put in the key of death which is apparently E flat? As my department members were listening into our conversation they were enthralled with what these kids were doing. They were connecting Lady Macbeth’s descent into evil with music. You can hear the three distinct parts which they had mapped out at the beginning of class. It was amazing that I had kids come in on their off-hours to finish the song -and better yet, these were all boys! Boys who were asking to continue the learning. One even asked me at the end if we could do this for every book we read. Tomorrow we are going to play it for the class, make changes, and add the lyrics and visuals.
I made a Photostory of the period 2 composition. I will also upload the orginal files here for period 2 and for period 5 of FinaleNotepad when they are completed. Here is an MP3 of period 2.
Additionally, here is a link to the webalbum of photos of their working. Check out their engagement!
Mapping of song in measures:
1-9 Lady Macbeth excited to see her husband return: happy, joyful, anxious
“Great Glamis! Worth Cawdor!
Greater than both, by the all-hail hereafter!
They letters have transported me beyond
This ignorant present, and I feel now
The future in the instant “(1.5. 61-65).
10-15 Decrescendo: Lady Macbeth explains her plan to Macbeth to kill King Duncan
“Glamis thou art, and Cawdor; and shalt be
What thou are promised: yet do I fear thy nature;
It is too full o’ the milk of human kindness
To catch the nearest way: (1.5.15-18).
16-21 Decrescendo to psycho Lady Macbeth divided into happiness at the prospect of becoming queen (16-18) and as the timpani enters her descending into madness (18-21)
“ I have given suck, and know
How tender ‘tis to love the babe that milks me.
I would, while it was smiling in my face,
Have plucked my nipple from his boneless gums
And dashed the brains out, had I sworn as you
To have done this” ( 1.7.62-67).
Period 5: Witches
Period 5 approached the song writing a little differently. We talked about who they wanted to write a song about and they chose the witches from Macbeth. With the witches, we talked about musically how you would communicate what each of the witches is all about. They came up with three separate instruments making up the sound distinctively of each witch, but how when they come together, they sound somewhat harmonious. They discussed that the witches symbolize chaos, confusion, unsettling, anxious, and ugliness. Shakespeare wrote the witches speeches in a rhyming pattern and the kids decided that they needed this to carry over into their composition. We also mapped out the song into a three part format where the witches represent the past, present, and future for Macbeth. So, in composing the song, they are going to use the past in connection to the line from scene 1, “Fair is foul and foul is fair.” The next part of the song captures Macbeth’s prophecies from the witches: Thane of Glamis, Thane of Cawdor, and then King. Finally, their song ends with the future, which is the witches’ desire to destroy Macbeth.
Overall reflection:
What was really interesting about the different approaches between my period 2 and period 5 is that it reflected how they have approached learning challenges in the class so far. Period 2 approached it collaboratively all working together, where period 5 broke into groups of music creators and lyrics composers. But with each group, I participated in awe of what they were doing. I felt so honored that they would do such exceptional work for me and with me. I truly feel so blessed to have been part of such an amazing experience. And then, to have kids thank me for giving them the opportunity brought home the reality of education. Why aren’t we extending more out of the box learning opportunities for our students? Why can’t them creating music to demonstrate their understanding of a character be just as good as writing a paper? Or completing a lab? Imagine if student had to write a song that demonstrates a chemical reaction? Or that expresses the emotions of soldiers dealing with returning home after serving in Vietnam. I really am so thankful to have been part of what my period 2 and 5 created. I hope the kids feel the same way. I know those who have expressed their opinions openly in class were so appreciative and enthusiastic. What a change in learning!
269 comments:
«Oldest ‹Older 201 – 269 of 269Mr. Fisch:
I think that we learn mathematics because everyone uses it every day. Whether someone is writing a check, paying for lunch, taking a class, walking, running, counting calories, or counting the pages of an essay, we all use mathematics. You'd have to live under a rock not to use mathematics. Literally, you would have to live under a rock! So, we are taught math because it is so amazing. Just by learning math, we can see how things work, from little toys to NASA spaceships.
I cannot explain how 2+2=D, all of the above, but I do know it is possible. I could explain it if I did about an hour of pure research, with no distractions whatsoever. But, we're in English right now, so... It's not going to happen. I think that some of the other students had completely awesome responses to this question, though.
We learn mathematics to find answers. Benjamin Peirce said that math was "the science that draws necessary conclusions". The reason we learn math is to get to the conclusion or answers of the questions we have. As for why the answer is D, see Logan.
So 2+2= 5, is this kind of like the thing in 1984? That's the main thing I remember from that play and how they were trying to brainwash the people. Also they were trying to make the people follow what there were saying and agree with what there saying by pain and by fear.
We learn mathematics because we use it in our everyday lives. Without learning math we wouldn't have the skills we need for our everyday lives. I seriously don't believe life would be as it is today without math. That's why we need to learn it. I have no idea how 2+2= D all of the above. Just look at Logan's answer. I still think 2+2=4.
By this time, I'd decided that I don't want to answer the 2 plus 2 question. I'm sure everyone can find some way to argue that their answer is right, and frankly, I just use the good old 'this is how you add' idea. I find it insignificant to know whatever the complicated answer to two plus two is, because when in the world am I going to have to care that two plus two equals ten. It was interesting, but I only register what I need in life, and a ton of stuff that really doesn't have any use at all in anything. That sounds a bit whiney, but that's what I say. Yeah. I really don't know why we do learn math honestly, or more, I don't want to think about it because I got less than 9 hours of sleep all weekend and thinking plus sleep deprivition equals really sucky brain function. There is no arguing that. In my professionally tired opinion, we learn math so that we can spit out complicated formulas and sound smart. Yep. There you go.
Mr. Fisch-
We learn math to be horrinble tortured in math class. haha not really. We learn math so that we can count, so we can tell what the quantity is. If you think about it math is very important. When Smitty handed out numbers how would we know what to do if we had never learned how to count? seriously numbers are like the bomb.
Okay, in response to the other question. I have no clue, I read logan's blog and I was so lost.
Logan- mind explaining that to me again? haha lo you can talk to me later ;)
9.99 reapeating=10
if x=.99 repeating then 10x=9.99 repeating.
if you subtract 1x you get 9x=9 and then divide by nine so x=1 so you multiply both sides by 10 and get x(or 9.99 repeating)=10.
(just a random thought from math class last year).
Mr. Fisch-
Well, I don't know how you got all of the above for 2+2 (Hey, I'm only in Algebra 1, so shoot me). And, after reading Logan's answer I am lost. So, it you need to know, see Logan's answer and then tell me how to do it!
In response to why we learn Mathematics and why it is important, I would have to start out with a basic "Well, we use it everyday". Math is necessary for life in our modern societey. How much gas we have, how much time is left, and how many minutes I have left. It is like that show on CBS, Numbers, where theye use numbers (Or Math)to solve crimes. We use it all the time.
John.
Mr. Fisch - I think that the purpose of math is to understand how things are made and are destoyed in the world. Learning is all about gaining wisdom and falling away from innocence that we are born with. The purpose of life is gain wisdom so that we may live, if that makes any sense.
I think that 3+3 can equal all of the above. For example, if there are three families, each consisting of 5 people, then I might say that 3+3=30. Some others may argue that 3+3=33 because if you put together the two numbers, it does equal 33. I guess that it all depends on perspective. We are all different so we see the world - and math - differently.
Karl,
We learn mathematics because we use math everyday, for everything. Math's purpose is to give us a sense of time. Humans today base so much of their everyday lives on time. On hours, minutes and seconds. A mere five minutes can truly change everything.
And 2+2=All of the above does not make sense. If you have 2 apples, and someone else gives you 2 more, then you have 4 apples.
Mr. Fisch
Not only does math teach us numbers, but it also teaches problem solving. We learn how to divide things evenly to avoid problems. Like in Lord of the Flies, if they would've rotated shifts watching the fire and hunting, things would have worked out better. And as for your answer to 2+2, I answered that it is 4. I think what you're trying to get at is that there can be room for interpretation and creativity with mathematics. Like in your story, it was mathematically correct to use four trucks, but three trucks worked better. 2+2 is whatever you need it to be.
What do you think? any responses to my answer?
Mr. Fisch,
Mathmatics is useful in many ways. The industrial part of mathmatics is very interesting, we couldn't live without industries and idustries use math. Think of all of the robots and new technology that math is creating. It's purpose is for the future. I will have to say that Logan has the best explanation, he uderstands this more than I do, I answered A. So you'll have to look at his. Thanks.
Laura
Karl-
Math is very important to us today. Math is important for so many different jobs and it's a skill that I think, as technology advances, that we need to know or we'll be left in the dust. As for answers to math problems I think that for some types of questions there are more then one answer. I think that if there is one answer there are many ways to somve the problem.
We learn mathematics because math surrounds us every day. The bridge you are crossing, the chair you bare sitting in, and in order to figure out what the discount is on 30% off, you need to be able to understand math. In order to function correct we need to be able to understand the world around us, and that world is full of math.
Logan made a good point with his answer to what 2+2 is and the more I think about Logan’s response, the more I agree with him.
Kailyn-what you're talking about is binary code. It's a way to program computers by giving only two options for a pixel, on or off. 1 or 0. So 4 in binary would be "100" I think.
Mr. Fisch- We learn Math because it's a method of thought. It helps the brain adjust to a certain chain of thought and reason. For instance, 2+2=x (in case of controversy). 2 is a quantity, plus means to give more. So we have to add the quantity 2 to a set amount of 2.
That wasn't a great explanation, but I think I got my point across. We learn math to learn to see things with a chain of reason to come to a conclusion.
For the answer D to his question, There can be so many variants in the chain of logic.
Besides, there are many ways to see a problem. We assume the quantity 2 is in base 10 (where 0-9 are the only digits), but there are other bases. By "base" I mean the number of digits one can use to express an amount. Like a said before in binary (base-2) 0 is 0, 1 is 1, 10 is 2, 11 is 3, 100 is 4, and 101 is 5.
every number is a combination of 1 and 0 in an increasing order to translate into base 10.
So in the chain of logic I mentioned before, the quantity 2 can be seen in different ways.
so in binary, 2+2=4 would translate into 10+10=100.
Let me know if anyone didn't get a word I just said/wrote.
"Why do we learn Mathematics? What is the purpose?"
Mr. Fisch posed this question after our discusion of 2+2 and global warming.
I think that the purpose of learning Mathematics is to teach us how to problem solve not only with numbers but in scanarios and situations that we can encounter in real life. Some of us, like Mr. Fisch, will go on to use the more advanced math later in life, whereas many will just use the basics. The important part of math is not to remember pi out to the twentieth decimal when you're eighty, but to be able to confidently approach problems we encounter.
The purpose of learning Mathematics is to explain what happens in the world around us. Its saying that 2+2 could be anything depending on the kind of math you are performing. The answer could always be different depending on what symbols you use to do the problem. Now i don't know calculus but it seems that in it there are a lot of different symbols so 2+2 then something besides = could make the answer something completely different than four. I think the purpose of math is that it will teach us that we can make things bigger with things that when only added together make something bigger, but when you apply several symbols then it can make a giant amazing outcome. The same thing applies with writing in Language arts. If you are writing an essay and you only use basic things and apply very little of the things you have available it wont be as great a piece. When you apply many things to the piece just as applying more signs to and equation it becomes bigger and better and you can get the same outcome of the essay. Math and Language arts are actually alike in very many ways.
Fish:
Silly? MIT is a tough school. Did I know? I think so. Math isn’t truthfully too complicated, but based on your ability to comprehend and memorize.2+2 = 4. I guess I haven’t been to college yet. However Logan explains it. Reasons to learn math generally are all together the same answer. We want to be the best and not 23rd. Education is not as usual in today’s world when the students don’t have the desire to learn more and better themselves with the opportunities given. You, a teacher and students have taken your opportunities to the next level. I hope we all can improve to the next level.
Students always seem to question why we need mathmatics in our daily learning, complaining all the time that they'll "never need it when they get older", but the truth is, we will use it every day of our lives. Math comes into play in our life every day in every shape and form. Writing a check and keeping track of it in our check books requires math. If we didn't know how to write checks, how might we buy our groceries and keep other checks from bouncing?
Math keeps us going through every day life.
I don't know how the answer is D, though, so "see Logan".
When I asked earlier for a mathmatical equation, I don't think you can have an equation with other values thrown in it without the problem stating you can do so. It's not asking for a different base, Binary, or different units. The problem asks for 2+2, and you cannot add anything to that to vary it. You can't change the value of the numbers or the operation or anything else.
If 2+2 really isn't 4, then the whole world of math is destroyed. Nothing will be as it seems and numbers won't mean anything. I do believe math is important in life and you use it daily, and defying it would mess everything up really bad.
I'm not the smartest math-wise, but I think we should stick to the principles of math. 2+2=? is asking for two and two put together, without changing it. You can't stick something else before the equals sign, because that makes it a totally different problem.
Mister Karl-i agree with ericak2012 when she says "By adding 2 and 2 and having it equal 3, you had to divide a part too." she just didn't explain it in a very understandable way.
The purpose to learning math is because we use it every day in even the most basic of life.
Even though some math is way o advanced to be used everyday in life by everyone, we still need to be able to know this so that some processes can be done. Also, without math, we wouldn't have sturdy buildings, because we would have no architects.
Mathematics, from the Greek mathematikos, coming from manthanein, to learn.
Alternatively, from the Greek mathema,meaning learning, study, science.
I would suggest that 2+2, stated as such, is pretty meaningless. It has no meaningful context. All of the answers I - and Sylvia - have given, are correct. And arguments could be made for more answers . . .
I'd like you guys to think beyond arithmetic, and think about the broader mathematics. Arithmetic is one very small part of mathematics. Beyond arithmetic, why do we make you guys learn mathematics?
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
- Albert Einstein
Oh, still waiting for someone to address why they think we use base 10 . . .
Logan's explanation for other bases was very good. For those of you struggling with it, think back to when you learned about place value. In base 10, the number '12' means one ten and two ones. But in base 4, the number '12' means one four and two ones, or the equivalent of 6 in base 10.
Therefore, 2+2 in base 3 = 11 in base 3, one three and one one. You can't understand mathematics if you don't truly understand place value (and different bases).
robertc2012 -
Making sure that a mathematical expression means something real is not defiance.
Of course we make assumptions about what 2+2 is. If your grandmother asked you that, you just say 4. You don't want to sass her and say, "two of what?" or "are you assuming a linear base 10 system?" You know what she means, so you just give the "right" answer.
But it's not the "mathematically correct" answer. The mathematically correct answer is that there is not enough information to solve the problem. If you don't define units, you may end up literally trying to combine apples and oranges. If you don't define direction, like in my explanation of the answer zero, you could end up going in circles.
Here's another example. You shoot a gun in the air, straight up. The bullet travels two miles (ok, it's a magic gun) and then travels two miles more.
If you believe that 2+2 equals 4, meaning the bullet will land 4 miles away, stay where you are. If you are starting to be convinced that in this particular case, 2+2 might end up right back on top of your head, zero miles from where it started, RUN!
2+2=4
2h + 2h = 4h
2 humans + 2 humans = 4 humans
So far, many of you - based on what you've been writing, and based on the "rules" of arithmetic you've learned - would agree.
(tristan and max) + (alexandria and hannah) = (stephanie and austin and john and marie)
My guess is that most of you would disagree with the last one. Yet the last one is a much more practical attempt at application of arithmetic than the first three.
Please modify that last comment. There's nowhere near enough information to be "practical" or meaningful in that example, but that example is more likely to be part of a meaningful application.
I think we use base ten, because the digits 0 - 9 and ten digits and make up every other number. We learn mathematics so that we can understand concepts that are man made and don't definitely exist. We learn it to get in the mind frame of things like time, things we invent.
Mr Fisch:
I think children learn mathematics beacuase, we need to understand the arthimitic part oe math. In other words, their is no point in learning something that you do not understand.
We learn mathematics so we can prepare for the future. There are numbers in everything, and the more we learn about them the more we can understand everything. But this is a difficult question. I don't think that anyone can answer this, even Mr. Fisch, can you answer this Mr. Fisch?
We learn mathematics because it is useful for many things. We use algebraic equations, geometry, trigonometry and arithmetics for everyday things without knowing.
For example, when my Dad and I built a desk for my brother's room, we had to keep the surface at an even 180 degrees while keeping the table legs at a 90 degree angle to the surface. This is geometry, but we probably don't think about it when we do it. Mathematics is essential in our lives, whether we know it or not.
Um, I'm very confused.
I'm still having a hard time with how in base 3 2+2= 11 or whatever. That makes NO sense to me. And base 4 just confuses me more. Can someone try explaining it again? I've read all of the previous explanations but it still makes no sense.
And I think we learn mathematics for some reason becasue they wouldn't teach us for no reason. I'm still trying to figure out the reason myself.
The schools teach mathematics because they want you to know the "science" behind everything.
There are many reasons why we have to learn mathematics. It teaches us about life. Mathematics will follow us around forever, whether we like it or not. Teachers have to prepare us for life, and this is the only way to prepare for life. Mathematics is one of the most important ways to do this.
There is no right answer. We can debate about this over and over again. For any given answer for 2+2, someone is able to come up with a possible way for that to be the answer. There are unending possibilities. It all comes down to the context.
So, the answer to 2+2= anything and everything
You make us learn mathematics because it is important to develop that part of our brains. It helps us to problem solve and think critically. The link you posted said that the word mathematics means to learn, so therefore we are taught mathematics to learn and grow.
I think that we need to learn mathematics because math elates to everything. Relating back to the story of the WWSC, even when you don’t have to think about math it still relates back to almost everything you do in life. If someone notices one small detail and then turned it into a solution, it is very likely that the solution was reached using math (in one way or the other).
What is the difference between math and mathematics?
First of all, I have no idea why we use math. But I do have to add one thing about the 2+2 thing. I think units are very important when talking about these problems. It could be 2ounces+2quarts, and that does not equal 4.
I got lost in this conversation when Mr. Fisch said that the answer is D. Can someone explain this to me?
Why do we learn math? We learn to math to help us cope in our everyday lives. If we cannot tell time or add the amount of time left on the timer on the oven we CAN survive. However, we cannot thrive. Our food will be burnt. We will be late. We might sleep too little. We might sleep too much. Whether we like it or not, there will always be math in our lives. There are shapes in stop signs. There are numbered pages in books. Math, clearly there is math, how much it relates to other subjects is the question to be debated.
To be honest I think one major resons we learn math is to apply it to science. Science is basicly real life. Every single one of our problems in the world today can be solved with science because science is real life. However math and science are closly linked so we learn math so we can learn and use science to solve the problems of our everyday sosiety.
I think that this is a really hard question to answer…
I guess that I’ve never really thought about it, I just knew we had to. But I’m just trying to think of why and the only thing that comes to mind is that we use it to think more deeply. I mean, really, if someone really likes math stuff, then they would find connections to it in the world and use it every day. Or, even if you don’t greatly enjoy all of that stuff, but you’re still good at it (that’s me) but I still find connections to it in the world. I’ll just be sitting in the car, bored as can be, and I’ll look at the clock and then I’ll start timing how long it takes us and then the average speed we probably went. But I don’t know, it just sort of happens…
We learn mathematics because…. Well I guess because it is something that is measurable and it adds value to things. We don’t learn math just to know that 2+2 = 4, because if it was only that than that would have no value at all. We learn math so that when we have two cupcakes and two cookies, we know that all together we have four deserts. It is applied to everyday life, because without mathematics trade would not work, money would not be worth anything, there just wouldn’t be any value to anything. So we have to know mathematics so that we can apply it to everyday life, where there have to be set values.
I agree with Bridget. Depending on the type of thinker the person is they can agree or disagree with any answers and have perfectly good support for their answer too. One answer may make more sense to someone, but that doesn't mean the others are wrong or even right. Who knows, this could go on forever.
I agree with bridget. There is always a way to make something what you want. I'm only in geometry myself but there is always some underlying foundation of the problem that can be altered to pu must first alter the foundation.roduce the desired answer. If you want to cange the shape of a building then yo
In preschool and elementary we learn the the bacics of addition, subtraction and mltiplication. However, we miss the real ideas and "philosophies" if you please. Mathematics reach far ther than these simple equation and problems of additon but having the ideas that you learn, being applicable to the situations that you encounter in every day life. I know I sound redundant, since I sort of said this in my previous comment but its not just arithmetic but the prinicples you learn. This is so different for me to explain because I never really have thought deeper into this than just the surface of the things I learn in school. Most of my teachers through my education have always bipassed the real fundamentals of it. I felt like they just focused on the grade you got at the end of the semester. Back to mathematics, to be totally honeset, I'm not completely sure. Basically in the simpliest terms I can say is that its learning how to learn and teach your self ideas and new ways to solve the problems that arise.
Anna, I thought that math was an abbreviation for mathematics,I don't think there is a difference!
Karl,
You make us lean mathematics because learning is a necessary experience, learning mathematics gives students experience learning. The better you can learn the more successful you will be in life. The only real job teachers have is to prepare their students for life outside of school and being a great learner is the key to doing well outside of school.
I was thinking more on the topic of mathematics and thought I would look up some quotes… Also, I thought that I may get a better understanding of how 2+2 has more answers than just 4. This is what I found (note: I did not put my interpretations of all the quotes, only a few):
Aristotle – “To Thales the primary question was not what do we know, but how do we know it.” It doesn’t matter so much as what the answer is, it matters how the answer was found.
Aristotle – “We cannot ... prove geometrical truths by arithmetic.”
Heri Bergson – “One can always reason with reason.”
George Cantor – “The essence of mathematics is its freedom.” There are many different ways for math to be understood.
George Cantor – “In mathematics the art of proposing a question must be held of higher value than solving it.”
Karl,
To answer this question I think that I must define what mathematics means to me first. I believe that mathematics is many different methods for solving multiple problems. We are some information and are expected to fill in the gaps to find the answer. So to answer the question why we learn mathematics, I think that it is not just for the obvious reasons like being able to add and subtract number for we which we learn math. Furthermore, I do not even think that it is really about the numbers at all. It is just practice solving problems in a myriad of different ways; for example, global warming!
More food for thought from early 2007.
@chelseac2012 - very interesting quotes. When you put them all together, what does that tell you about mathematics?
@kailynw2012 - thanks for taking a shot at why we use base 10. The digits 0 through 9 certainly add up to 10 digits, but only in base 10. Kind of a circular explanation, there. If we used base 6, we'd have 10 digits as well (stay with me here, 10 in base 6 is the same as 6 in base 10), 0 through 5, so that argument would hold up just as well.
Hey everyone,
It looks like you have a nice conversation going about Macbeth and all but I got lost when the whole 2+ 2 thing came into the conversaition. Anyhow, hope that you enjoy Macbeth with Mrs. Smith!
We learn mathematics not only so we can learn numbers and how they work, but to understand life and how to work with it to come up with what is perceived to be the best solution to each of the decisions we make. With each new decision there becomes a new problem and with each correct answer we get the clearer everything becomes. For an example if someone were to ask if you wanted a cigarette and you said, “No thanks I would rather not.” Then your path becomes clearer and you would know that you have a better chance of not having lung cancer compared to someone who would have said yes to the person who had offered them the cigarette.
I also agree with Justin when it comes to the 2+2= D) all of the above.
Mr. Fisch has done an excellent job of getting everyone to start thinking about critical numeracy. There is a saying "numbers don't lie", or do they? There is another saying "50% of all statistics are made up on the spot." You can twist numbers to say what ever you want them to. Being able to critically think about numbers and statistics is an important life skill.
Instead of chiming in with another answer to the 2 + 2 problem, I will pose a thought. Maybe what you think of as the question is the answer, and what you are really trying to figure out is "what is the question." 2 + 2 is only one side of the equation. Does the "answer" on the right side of the equals sign have to be a whole number?
Mr. Zack Allen-
I think you pose a wonderful question! Of course, that is a wonderful way to look at things. I think I know what you may be talking about. In simple terms, here is what I think you mean:
Answer : 4
Equations: 2+2, 2X2, 6-2, 10x0.4, And many other equations. Thank you for the great opportunity to look at things a completely different way!
And, for you second question, no I don't believe that the answer has to be a whole number. I think the others' examples with whole numbers was purely because it is an easier way to start understanding these "there is more than one answer" problems. I believe there are zillions (not a real number, I know!) of answers to the simple question, 2+2.
Again, thanks!
Mr. Fisch – From the quotes, I drew that the answer isn’t really what matters in a problem (whether it be mathematics or not), it’s how the answer was found. From the quote, “One can always reason with reason.” One can always think of another comeback for what someone else says. People have different opinions; in math, the reasoning people have for their answers, anyone can come back and have a different reasoning to contradict. Different people understand mathematics in different ways.
I'll take a shot at why we use base 10. We have 10 fingers. It is natural to start "regrouping" when you run out of fingers to count on.
Similar variations have been used by the majority of cultures throughout history. The Maya used 20 (fingers and toes), many cultures used 5 (one hand), and some used 8 (no thumbs). The Sumerians used base 60. I have no idea what they were thinking. I'm just glad we moved past that.
Ben,
Check out how one counts to 100 in French.
So, if it's because we have 10 fingers, then there's nothing magical about base 10. If humans happened to have 12 fingers, do you think we might be using base 12?
(Except, of course, we'd still write it as 10 if we had developed the same naming conventions, we'd just have a new single digit symbol for our current 10 and 11, much like hexadecimal - base 16 - uses A for 10, B for 11, etc.)
Gary: Wow. I am glad you mentioned that to me, because frankly I had no idea that's how it worked. For the rest, here is what I found. (Look where it says "notes.")
I suspect that a system like that evolves out of colloquialisms, and possibly a cultural tendency to break relatively common numbers up into more manageable chunks.
In response to Mr. Fisch, I think that we would probably count in base 12 (or 24 or 6) if we had 12 fingers. And we might be better for it. Doing a little more research after this topic initially got me thinking, I found that base 12 has been put in place, somewhat artificially, because it is easier to break into parts.
Also a YouTube video of counting to 100 in French.
Mr. Fisch- I really enjoyed your scenario about 2 + 2 = 3. It was creative. I couldn't help but think of the passage in "1984 when Winston is being tortured by O'Brian into believing that 2+2=5, which in this case represented the ability of the government to brain-wash people into believing untrue things. However, I do think the math class is basically pointless past fifth grade unless one plans to go into a field involving math, such as engineering or chemistry. I understand that we must learn simple multiplication/division and other elementary concepts in our day-to-day lives, but for anyone not planning to be a scientist or mathematician, math in no way benefits them in their careers or their lives. Personally, I have no use for it. I plan to be a dancer and an English teacher or novelist. None of this requirest that I know how to form a quadratic equation and graph it. Seriously people, people are starving, genocides are occuring, and people around the world can't afford a roof over their heads, there are more important things to learn than y = 8x + 6. And, the way math is taught in today's schools is corrupt. We take notes, do problems from the book, take a test, and take a final. That is the complete extent of our curriculum in math. No matter how fun the teacher is (I love my math teacher to death) there is no way you can convince me that my time in those classes are well-spent. This doesn't mean students should not do their best every day, but I just think we either need a radical change in the way we teach math, or we need to make it an optional thing at the high school level.
I think that math should be required simply for the reason that people are hungry and other large world problems we need those people that know math and science well enough to be able to go and change the world and solve major problems. Also, if you never are forced into math how will you know if you like it or how good you are at it? Just because you don't like something doesn't always mean you're bad at it. And I think that the U.S. needs more world class mathematicians and scientists to solve the world's problems and if noone tries to make it that far then noone ever will.
hannahl, I've been thinking about your comment for a couple of days. I think what Orwell was trying to show with Winston being tortured to believing that 2+2=5 was that the power of the government was absolute, that anything they said was true. It's sad that sometimes maths seems like that.
But what we've been talking about here is not demanding that one answer is right, but to explore what other answers might be right. The freedom to question, to ask impertinent questions, to explore unlikely answers is the key to fighting fascist behaviors. It's not about math, per se, it's about opening your mind to the world and seeing possibilities where no one else has before.
Mr Fisch- A great point about the fingers. In response I have to follow the style of the hitchikers guide to the galaxy.
If we based the world's most common system of logic and arithmetic on the completely random sequence of DNA that caused our race to develop 10 fingers as opposed to some other random quantity.
Did we base the world's main system of logical quantities on random DNA that affected us most billions of years ago and could change at any point as far as we know?
I know it's strange but it' worth considering
Gary- You bring up a great point. I took french for a while and I also speak portuguese fluently.
Both languages (Latin languages in general) use the same basic system for counting.
0-10 are different words
11-19 are variations of 1-9
and 20-99 are combinations of a variation of 1-9 (in the tens place) and the same words for 1-9 used in the first few digits (in the ones place)
After 100 a similar system continues with variations of the digits 1-9 for the hundreds place.
Sorry, I got cut off.
What I was trying to say was that if the number system was based of the number of fingers we have (you can do 5 for one hand, 10 for both, 8 eith no thumbs, etc.) why would all of these different places have the same way of counting?
The numbers must have originated from a particular place to be so similar. I think it was greece because they had contact with every civilization and could have spread the system.
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