exploring the quantitative world

Danya and Brittany Final Project Description

Vi Hart comes up with an incredibly intriguing way of thinking about math. Her videos are entertaining and captivating while illustrating a new method to think about the process of math. In short, her videos are a good encapsulation of everything we’ve been doing this semester.

Our project seeks to encompass all of the qualities that Vi Hart illustrates in her videos. One question we wanted to explore was how could we implement an interactive game or thing that would better explain or help students experience a math problem? Our main objective was to see how we could connect exploring a new idea and relate it to math. The biggest correlation between our project and Hart’s projects are the concept of doodling and drawing and their relationship to math. Rather than taking one specific video, we decided to look at a few of her videos as a whole and see what her bigger ideas were all about. She essentially makes math a recreational project. When we were initially brainstorming how we wanted to explore new methods of making sense of math and asking mathematical questions, one thing that drew us in was idea of drawing certain designs and measuring things like their areas or circumferences as well as other aspects like number of lines produced within a design based on the number of levels implemented. We ultimately designed a project where we could implement those characteristics. We landed on creating three different sprites that illustrated different designs, respective to their shapes. The way we connected math in this case was we wanted to see what the relationship is between levels inputted and number of lines illustrated in the final design. It was obvious to see that many of the designs drawn were symmetrical, and we wanted to see how much of a connection there was. We discovered that the relationship is exponential in the end. In class, we will illustrate this phenomenon by showing the project on the projector for everyone to see our methods.

            This class so far has successfully introduced a new way of thinking about math. For the two of us, we never fully embraced the process of math in a positive light. Taking this class opened both of our minds up to a different methods as well as introduced us to the idea that math does not always need to be seen in a negative light. One of the most important things we both learned through the process of programming was that exploration can lead to a greater and more thorough understanding of this process as a whole. 

Ch.8 Mindstorms

I am glad that Papert finally addressed the connection between a LOGO environment and schools.  I have really enjoyed reading Mindstorms and found Ch.8 to be my favorite chapter to read.  I liked is comparison of a LOGO environment to the samba schools.  To him, the samba schools were his mathland where math is a real activity and novices and experts are learning together.  This type of environment reminds me of a Montessori classroom where students of different levels are learning from one another while engaging in different activities.  The role of the teacher is to get the students started on the activity and to guide them if they need it, but otherwise the student is mainly in charge of their learning.
I definitely agree that math is cultural and that the computer can be used to unite multiple domains.  But what is really important is the process, not only of mathematics but of other subjects too.  I think that is why computer programming is such a powerful too because the skills that are learned through it are important to all areas.  Computer programming also gets away from the idea that there is only one answer.  Plus the process of debugging is just as powerful as programming.  

In order to implement a LOGO environment or even to fully integrate computers into the classroom there needs to be a complete redoing of the math curriculum.  I liked Papert’s comparison to the change of how skiing is taught and learned.  In order to change math in school the content, pedagogy, and technology all need to be reformed.  Unfortunately, I think we are a long way from this reform ever happening and that by the time the math curriculum is reformed, there will  be something even better than computers with which you can teach math.  However, I think until there is a reform of the math curriculum, computers can be used enhance the current curriculum.  I think the biggest hurdle in fully implementing computers is the state math standards and standardized tests.  Since there is a set math curriculum for schools, there is little room for students to explore math and other domains with computers.

Mindstorms Chapter 8 Response

            In Chapter 8 of Mindstorms, Papert finally answers the question that he has been building towards for the entire book: what is the ideal social context in which his type learning might take place? Up until this point, Papert has emphasized and reemphasized his computer culture that moves away from the traditional mathematic one, allowing us to learn (in the traditional sense) as well as “learn about learning” (177). The culture that he has described, to this point, has incorporated humanized learning with almost personal relationships with knowledge, allowing the learner to supplement their experience with other students and teachers. Papert addresses the idea that whenever talk of schools ceasing to exist is brought up, people are naturally inclined to go on the defensive, as “most of us spent a larger fraction of our lives going to school than we care to think about… the concept of a world without school is highly dissonant with out experiences of our own lives” (177). Despite, he does suggest that if we are to move in this direction, it is vital that we begin to plan “elaborated models of the nonschool activities in which children would engage” (178).

            I felt that the most intriguing aspect of Chapter 8 was Papert’s subsequent discussion of Brazilian samba schools as an example of such a model. As he explains,

“Every American disco is a place for learning as well as for dancing. But the samba schools are very different. There is a greater social cohesion, a sense of belonging to a group, and a sense of common purpose. Much of the teaching, although it takes place in a natural environment, is deliberate… and focused. Then it dissolved into the crowd.” (178-179)

Papert then astutely connects Brazilian samba schools back to Turtle Geometry and the LOGO environment, claiming that they are alike due to the fact that any mathematics associated with either samba school or Turtle Geomety is “a real activity that can be shared by novices and experts. The activity is so varied, so discovery-rich, that even in the first day of program, the student may do something that is new and exciting to the teacher” (179). Furthermore, he elaborates on this by bringing up yet another similarity: the quality of the human relationships each creates. The way in which teachers intervene in learning in the setting of Turtle Geometry mirrors the way in which expert dancers in samba school assist and learn from others. Instead of having a traditional curriculum or lesson plan, teachers working with Turtle Geometry answer questions, provide help, and facilitate students by providing insight that will help to expand their horizons. Similarly, expert samba dancers will help to assist those who are struggling, answer questions, and – just by dancing – allow others to visually model what they need to do. Additional, Papert describes how the samba school and Turtle Geometry share a third thing in common: “the fact that the flow of ideas and even of instructions is not a one-way-street” (180). In Papert’s own words:

“The environment is designed to foster richer and deeper interactions than are commonly seen in schools today in connection with anything mathematical. Children see programs that produce pleasing graphics, funny pictures, sound effects, music, and computer jokes. They start interacting mathematically because the product of their mathematical work belongs to them and belongs to real life. Part of the fun is sharing, posting graphics on the walls, modifying and experimenting with each other’s work, and bringing the ‘new’ products back to the original inventors.” (180)

The only thing standing in the way of such a learning environment and the subsequent growth of popular computer cultures, according to Papert, is a cultural obstacle. The remedy, therefore, is also cultural, and thus Papert hints that in order to advance his model learning society, we must “advance the art of meshing computers with cultures so that they serve to unite… the fragment subcultures that coexist counterproductively in contemporary society” (183).

Minsky and Mindstorms Ch. 7

I took Intro to Cognitive and Brain Science last year, and in that class we were introduced to the concept of the “self” being an oversimplified illusion, and not displaying the complexity of what is really going on in the brain. Having been raised with the language of Eastern religions often around me, such as Self with a capital S, this was an interesting thing to ponder, and still is. In that class we also had a couple of lectures on Artificial Intelligence. Minsky’s article discussed the two together (selfhood and AI). When I heard the lecture on AI in the CBS class, I remember being struck by how abstract, “math-y”, and boring the whole thing was presented as. I liked the first three minutes of the lecture, because it introduced how computers could be used to understand human minds, and vice versa in some cases. But then the lecturer went on to introduce abstract formulas and concepts that were totally divorced from things I could relate to. This experience connects directly to Mindstorms Chapter 7, where Papert is emphasizing that studying how one learns is just as important to learning what one learns—and that the two are not separate, but that actually, as shown in the “learning to ride a bike” example”, understanding the process can help you achieve the skill. In trying to teach AI, the lecture failed to see how the process of “accessing” AI is just as important as understanding the different algorithms and formulas that allow for AI programming. 

Something I want to bring up is what I see to be great hypocrisy in both of these articles in how they seem to view science as a more valid form of learning than other fields. Minsky writes, “so ‘consciousness’ yields just as sketchy, simplified mind model, suitable only for practical and social uses, but not fine-grained enough for scientific work” (Minsky 15), while Papert writes “By restating Newton’s laws as assertions about how particles…communicate with one another, we give it a handle that can be more easily grabbed by a child or a poet” (Paper 172). There is clearly this concept that science is “real” in a way that a poet or a social scientist is not. While I understand that Minsky is arguing that we are deceived by our daily experiences of selfhood, the reality is that this is what we experience. He himself says that it would be too risky to suggest seeing humans as something other than selves, given how our societies are so structured around this concept. So those “lowly disciplines” that take the experience of selfhood into account, while perhaps are not getting at the reality on the neuronal level of processing, are getting at the reality of experience. And to invalidate this type of knowledge as lesser, as their language of scientific-centrism does, goes against Papert’s whole argument about making math more accessible and meaningful. While perhaps it is not contradictory, if he really means that math should be packaged so that less robustly scientifically intellectuals can still make sense of it, I have always gotten the sense that he is more talking about showing the interconnectedness of knowledge. And so if he is against the fragmentation of knowledge into supposedly separate disciplines, why the need to locate science as this separate body of knowledge that is more “real” than other forms?

The line in Mindstorms chapter 7, “the dynamics of lift are fundamental to flight as such, whether the flyers are of flesh and blood or of metal” (Papert 171) certainly connects to the Minsky article, because Minsky is trying to lessen the gap between human brains and computers, and show that they don’t have to be seen as so fundamentally different. Here too, Minsky is emphasizing the process of flying, versus the body that is doing the flying. I think part of why this is a hard concept to grasp is because of our “self-oriented” mindset. Socially and culturally, we rely so much on the idea of an individual and distinct self that we place great focus on “who” is doing the action. Minsky is saying that the “who” is less important, in that the human brain isn’t so different from the computer brain, and the processes are similar. And Papert is saying that the process shouldn’t be separated from the end result—that the process can be illuminating, in fact.

Mindstorms and Minsky

So I replied to Nick’s e-mail about meeting up to video chat, but I later found, in my spam folder, an e-mail saying the message was not sent (there was a problem because I didn’t use the e-mail address that is on the list for the turtle geometry google group list), so that didn’t work out. We talked about a lot of this in class already, so I will just write a little bit about what I think.

Reading these did actually change how I think about AI. I didn’t really have a clear definition of AI before. I just thought of it as being machines or computers that think like humans somehow, but I was not sure what that meant. Now, I think of AI being characterized basically by machines or programs that can learn things. This way, they can improve or adapt for future possible events as they gather more data and develop a history to help them predict future problems. 

In Mindstorms, the author mentions AI being a way computers emulate human intelligence. I think the idea of intelligence is hard to define, but I liked the example of people studying the flight of birds in order to create airplanes. Airplanes are machines that emulate the flight of real birds, so airplanes appear to have the intelligence of birds, even though they are machines made of metal. I think this concept of AI still needs to be developed and refined over time as people research more into this area and create new technologies. 

The author in Mindstorms also goes back to his point from earlier about breaking difficult concepts into microworlds (like the physics microworld) in order to learn them better. He also mentions that computers can help children construct their own microworlds, and he provides a few examples. I can definitely see how computers and Turtle Geometry can help people understand and practice this concept of breaking difficult problems into smaller, more understandable concepts. The use of functions or methods in programs demonstrates this very well, but it is also about thinking of how you would do things yourself in basic terms so that the computer will understand how to do them.

Mindstorms 7 + Minsky

Reading the Minsky article in conjunction with the mindstorms chapter made me think a lot about the computerized chess program, specifically “Deep Blue” which was a computer-chess player that ultimately won against the chess champion human. Before finding this out, my instincts were to say that a computer would never be able to beat the best human chess player because of a factor of human’s creativity, spontaneity, and ability to predict moves ahead of time and also predicting the other players moves by evaluating the way they’ve been playing. It’s a little startling that these conditions that I thought were unique to humans with alive and functioning brains are things that a computer is capable of doing. It brings into question of whether the ultimate goal of computers in a world of technological advancement, is to imitate and reproduce the workings of a human brain. Is that just creating humans, then? 

I still believe, because of personal experience, unlimited expanding knowledge, unpredictability, and interaction with similarly-capable brains, that humans maintain an upper-hand of creative capabilities and understanding over computerized systems. There is still the sense that we control the computers, that we make them do things, and that any sense of “personality” we try to instill in them, really is just a projection of human choices stemming from our unique capacities/traits. 

scratch

Here is my attempt at embedding triangles in another shape! 

Ruby Processing

Once I got ruby processing to work on my computer, I decided to give it a try.  I looked over some of the example codes in the Learning Processing with Ruby in order to get a sense of how it worked.  I ended up just messing around with the code and changing things in the code to figure out what the code actually meant.  

My program creates a square and then everytime you click, it creates another square that is larger than the previous square.  The program also changes the colors of the squares after each click so that the colors are getting lighter.  If you want to start over you can press any key and it will reset the size of the square to the original dimensions.  I really did not have a project in mind when I started working with ruby processing so this is just what I came up with.  

Here is the code:

class Squares < Processing::App

  def setup

    background 255

    smooth

    @rect_x=150

    @rect_y=150

    @rect_size =50

    @rect_fill = 50    

  end

  def draw

  stroke 0, 90, 0

  rect @rect_x, @rect_y, @rect_size, @rect_size

  fill 0, @rect_fill, 0

  end

  def mouse_pressed

    stroke 0

    fill 0,0,@rect_fill

    @rect_size +=10

    rect mouse_x, mouse_y, @rect_size, @rect_size

    @rect_fill +=20

  end

def key_pressed

    background 230, 0,0 

    @rect_size=50

  end

  end

Squares.new :title => “Squares”, :width => 500, :height => 500  

I was also still working on the triangle recursion from Turtle Geometry in BYOB, but when I create a function that calls itself, I cannot seem to get it to work.  I am going to continue to work on it and see if I can get it working.

Recursion in Scratch

http://scratch.mit.edu/projects/ngwoolf/2146678

Here’s my attempt at making a program to simulate recursion in Scratch. I based it off of a project made in scratch that used a similar method to make shapes and animals that kept growing… However, I couldn’t figure out an efficient way to make it keep repeating/growing forever (or at least, for more than 5 steps), as I could only get to 4 or 5 levels before it broke down. Furthermore, I’m still working to get it so the shapes are smaller and fit on the screen as well as to install a counter that allows you to adjust the level.