It's all about the language...

I had high hopes when I read the title for the first section of Eisenburg’s first paper: “the notion of a “transitional object”between concrete and formal reasoning.” Anyone who has shared a class with me knows that I am passionate about the use of physical materials to explain abstract concepts. However, this section was missing a key ingredient for me: how does the logic become formalized? Math is a language of symbols and abstract representations. In order to apply the concepts imbedded in these symbols, a student must also be fluent in the language of math, the symbols. In order for Mindstorms and other constructionist tools to be effective in truly teaching Math, it must not just teach concepts like patterns, velocity, etc., it must also teach students how to communicate these concepts in the world in which they are used.

I believe the true “transitional object” in logo is not the turtle itself, but rather the input language itself. While the turtle turning 4 times at a 90 degree angle and moving forward to create a square may show a child what a square is, it is the act of entering the code to move the turtle that forces the child to formalize the knowledge they have acquired. My research for my Masters Project showed me that children, especially young children, are not easily able to take concepts they have learned from physical objects and apply them symbolic representations. For instance, a child playing with fraction bars may recognize that 1/3 is bigger than 1/4 was not able to solve the same problem on paper. I think Logo provides a powerful example to how language may be used to provide the bridge between formalized knowledge and concrete concepts, however I believe Eisenburg misses this key ingredient of this formalized link.

I realize the manner in which I discuss these mathematical ideas is exactly the type of language that Edwards warns against. I often find myself discussing mathematical ideas as fixed pieces of knowledge to be transferred, rather than constructed. While I believe the discussion as to how we think about and value mathematical knowledge is important, I have chosen to pursue ways to help children learn mathematics within the system of symbols we have developed. Success in traditional math classes, developing quantitative reasoning, reaching Algebra, etc. is a determining factor for economic stability. It opens doors into new careers and can change the course of a person’s life. For this reason, I focus my attention on helping students construct meaning for themselves of the socially constructed symbols.

Apologizes for the tangent. However, I believe Edwards discussion of the importance of meaning imbedded in language is a worthy one. One thing Microworlds do beautifully is create a language for failure: debugging. In fact, failure is presupposed for learning within these environments.

“if the learners fully understood the mathematics or science they encountered in a microworld, the program would have little value.” –Edwards

In most of school, but especially in Math classes, success is defined as reaching the correct answer. A teachers value is often defined by how quickly he or she is able to get students to correct answer. There is little room for exploration in these environments. I worked in a private middle school in Manhattan where expectations for students were sky high. My students would soon be applying to prestigious high schools around the city and their tests scores were often seen as determining the rest of their future. These kids had so much pressure put on them, it was difficult to watch them struggle. In response to environments like the one I was a part of, many educational tools are designed to make learning easier, to take away the struggle. But, as Edwards points out, it is in the struggle where the learning happens. The work “de-bugging” implies that failure is normal and expected. If only we could bring this notion into school. (perhaps even elite Universities…)

Microworlds Code

Building a triangle of set size

to triangle

rt 30

repeat 3 

  [fd 100

   rt 120]

end

Building a house

to square

repeat 4

  [fd 100 

   lt 90]

end

to house

triangle

rt 150

square

end

Square of any size

to sq :size

repeat 4

 [fd :size 

   lt 90]

end

Polygon of any size

to polygon :side

repeat :side

  [fd 50

   rt 360 / :side

   ]

end 

House of any size

to tr :size

rt 30

repeat 3 

  [fd :size

   rt 120]

end

to hs :size

tr :size

rt 150

sq :size 

end

Circle of any Size 

to circle

repeat 360

  [fd 1

   rt 1] 

end

 Animation

Using the backpack set shapes to change from frog3 frog 4 frog 1 frog2….

 

Defining Red

First: draw red border

Second: right click on the red, go to edit red. Under turtle I typed the directions, rt 130, causing the turtle bounce around the circle

Two Turtles

to tree

fd 60

repeat 3

  [lt 100

   lt 30

     repeat 120 [fd 1 rt 2]

   ]

lt 155 

fd 60

end

to grass

repeat 3[lt 20

     fd 18

     lt 160

   fd 20

   rt 155

     fd 22

     lt 145

   fd 23

   rt 150

      fd 25

      lt 150

  fd 20

  rt 175]

repeat 3[lt 20

fd 18

     lt 160

   fd 20

   rt 155

     fd 22

     lt 145

   fd 23

   rt 150

      fd 25

      lt 155

  fd 20

  rt 170]

end

To Picture

cg 

jill, pu jack, pu

Jack, setpos [-40, 0]

Jill, setpos [100, 0]

jack, pd jack, tree

jill, pd jill, house

pu jack, pu

jill, fd 120, launch [jack, fd 120] 

jill, rt 180, launch [jack, rt 180]

jill, pd  jack, pd

jack, pd

jack, repeat 17 [grass]

A song!    (I bet you recognize it!)

to song

note 64 10

note 62 10

note 60 10

note 62 15

note 64 8

note 64 8

note 64 8

note 62 8

note 62 8

note 62 8

note 64 8

note 67 8

note 67 8

end

Reversing a list:

to reverse :list

        ifelse empty? :list

                [output []]

                [output fput (last :list) (reverse butlast :list)]

End

Button 1 and Reverse Output (see pic)

 

Button 2 –Song (see pic)

Microworld Pics

The Big Idea

What's the big idea?

"Technology often forces us to choose between quality and convenience." -Kay

This quote from the Kay reading sums up my view of educational technology quite well. When building tools for learning, developers often pick from the low hanging fruit, those easiest to asses and implement. As a result, computers and technology have been used, as Papert pointed out, not to revolutionize education, but to re-purpose old methods of education. The recent stimulus package passed by congress a great example of Papert’s assertion that education is often about immediate problem solving and not a holistic approach to education; how well intentioned problem solving addresses the symptoms, rather than the cause. Obama has finally provided school districts with large amounts of funding to reach the goals of No Child Left Behind, providing funding for exciting new technologies and professional development. However, these funds were provided with provisions that districts meet strict standardized test score benchmark set out by NCLB. What happens then, when a child enters the 6th grade at a 3rd grade math level? They are bombarded with kill and drill software that will help them succeed on standardized tests. But have they actually learned the material?. I find it ironic that Papert uses the success of one of his students on old tests to show the success of logo. Until we change the way that students and teachers are evaluated, educational technology will continue to be designed for the path of least resistance to success on small ideas.

“An educational system that tries to make everything easy and pleasurable will prevent much important learning from happening.” Kay

One of the reasons “kill and drill” software is so pervasive is that it harnesses the power of computers to quickly asses and respond to students. For instance, FastMath keeps tracks of how quickly students are able answer math facts and keeps asking the students problems they have trouble with. Imagine if computers were used instead to scaffold more nuanced problem solving. As the above quote by Kay alludes to, the key to learning is often in creating a good problem. Good problems can encourage students to find the big ideas that Papert refers to. However, I found these good problems the most challenging to generate. It is difficult for teachers to adapt to the varying needs of their students. Technology could provide a great helping hand in this arena.

Big Ideas...

What is a big idea? Too often in education we equate bigger and faster with better. Reading bigger and bigger books faster, memorizing more and more facts, counting higher and higher? Do big ideas always have to be more complex or advanced? Most of the Media Lab’s projects are designed at helping kids to access more advanced ideas. Even Papert’s 1971 paper on using computers in classroom, which provided exciting examples to bring engagement and discovery to the classroom, focused primarily on helping kids to understand advanced principles. While flashy programmable robots and kids learning Newtonian Physics are impressive, I found the example of child discovering zero, more significant to that child’s learning. Technology can not only give students access to understanding complex ideas like calculus, but can also help them grasp more fundamental concepts such as inquiry or addition.

“At the same time, the new technologies should provide children with conceptual leverage, enabling them to learn concepts that would have been difficult for them to learn in the past.” -Resnick

My issue with Logo…

While the examples of kids really embracing Logo, how do you inspire kids to put in the initial effort of learning the language upfront? Clearly, not all kids took to programming the same way Michael did. It seems like this document is just a manual to be copied, not on to help kids discover on one’s own. Examples are great! They help us learn. But how do you give examples to learn from but still allow for discovery and creativity. Can you give the tools to solve problems on their own without providing manual or almost identical examples?

Gears Essay: Jigsaw Puzzles

When I think back on my childhood, it is difficult for me to pick one item or one experience that influenced the way I think or the direction I headed. I was constantly trying new things and had new interests all the time. So, I went to the only person who knows my childhood even more than I do: My mother. She actually replied similarly. She told me that I had new projects all the time. However, she also told me that was constantly creating new challenges or myself; how fast can I race my big wheel down the driveway? How many baskets can I make in a row, etc. Then, she reminded me that while most of the kids were playing house and dolls in preschool, I was often in the corner devouring jigsaw puzzle after jigsaw puzzle. I think playing with puzzles developed my love of challenges and problem solving, allowing me to pursue lots of different interests and informs the way that I attack problems and challenges now in my adult life.

The thing I liked most about puzzles, is that you had a vision of the solution: the box top. If you only saw the pieces, you have no idea how exactly the final solution is supposed to look. However, the box top allowed a constant reference to scaffold to the solution. Puzzles taught me how to break apart a large problem into many smaller pieces. The first thing I would do when given a puzzle was to separate the edges from the pieces in the middle. In other words, attacking the simplest part of the problem first. This is similar to beginning a problem with what you know to be true or figuring out the easy things first. Next, I would look back at the box and decide what the easiest section to complete would be. These were usually the places in the picture that had the most distinct features. I would see which colors were prevalent in these sections and begin to search through the piles of pieces for the ones that could be used in these sections. I would repeat this process over and over, ending with the most complex piece of the puzzle. However, the more difficult parts were much easier to complete once the amount of pieces were limited.

I now try to approach problems in a similar formula. I often find it helpful when learning mathematical concepts to begin by looking at the solution to sample problems and figure out what it takes to get to the solution. I enjoy taking complex problems and breaking them apart into smaller and more manageable mini-problems. I have also found it helpful to focus more on what I do know, rather than what I am missing. If you continue to look at the large pile of pieces left in a puzzle, it is easy to get overwhelmed. I like to keep focused on the small part of the problem I am working on and reassess the over all problem in between each section. It is always exciting when a puzzle begins to look like the picture you are trying to create. In the same way, I always gain a lot of satisfaction when I have chipped away at a difficult problem enough to have the solution in sight.

In elementary school and middle school Math was always my favorite subject, because the problems themselves felt like puzzles. I loved attacking problems from different angles and breaking them apart to put them back together again. In 5^th grade, my teacher actually let one of my classmates and I form our own Math group and we taught ourselves in the hallway every math class. However, in 8^th grade, I moved schools and began having a very different experience in Math class. I was no longer allowed to progress at my own pace, but was forced to follow along in the book with the rest of my class. Math no longer felt like a puzzle, but a set of rules to be followed. To extend the analogy of the puzzle, it was like having someone stand over your shoulder and telling you where to put each of the pieces and in what order, rather than helping you to develop strategies on your own. So, while you may have completed that single puzzle you felt little sense of accomplishment or desire to take on a more challenging puzzle. Little, by little, I lost my love for mathematics.

However, I never lost my love for challenges and puzzles and eventually found my way back to mathematics. I became a Math Teacher in hopes of inspiring this love of problem solving in my students. I tried to help them view math, not as arbitrary rules about numbers, but as a way of approaching challenges. By building stronger conceptual understanding, I hoped they would develop not just math skills, but strategies as well.

I came to Stanford for two main purposes: to learn more about how children process mathematical ideas and to design tools to help them learn these ideas better. In doing so, I am attacking the giant problem of math education in the same way I would any puzzle. One small section at a time.

Welcome to April's Education236x Website!

Welcome to my website for 236x. I am a Master's Student in the Learning Design and Technology program in the School of Education. I am a former middle school math teacher and am interested in how technology can enrich student understand of mathematical concepts. Looking forward to a great quarter!