August 4, 2006

Reflection on The Algebra Project

Event

Materials: An assortment of household items, tools, gadgets(we found out later they are called mechanisms)

Mechanisms – scissors, tongs, hole punchers, eyelash curler,

We played, experimented with them, asked questions about them
What is this , How does it work, How are these alike, Do I have these in my house? I need one of these, etc.

We were asked to select one and use it for its intended purpose.
Our group selected the scissors and we started cutting paper. We cut single sheets of paper, then we folded the paper and cut them. We wanted to find out what made the scissors easier or harder to cut with. We were checking how we used of fingers, what was happening with the blade. There were several factors that we noticed that made a difference.





The width of the paper
Where we placed the paper against the scissors

We were thinking about the motor development skills of the children and how that would determine difficulty.

It helped when we drew a picture of the scissors to view the vertex( the angle where the two blades meet)

August 4, 2006

The past few days our group has been working through a second event in the Algebra Project. As our event came to a conclusion I heard the words that this event was a science focused event. This was a new revelation in my mind, as we had been working the past few days I didn’t think, “this is science.” Event 1 verses event 2 seemed to be very similar. Event 2 encapsulated some physics aspects; yet it seemed to resemble a great deal of math. As I look back on the steps we took through event 2, thinking as this event as a "science" event, I see the science aspects in the actions we took. As a teacher, I more have more understanding of how the algebra project can overlap with other subjects in my classroom. There is no clear cut line for when you are working with science or math or even social studies. The circle that illustrates our process felt exactly the same as we worked with science geared information as well as math. I worked through each event, finding myself driven my curiosity for what knowledge we were going to come across. The “revelations” at the end of the events were very meaningful for me. I found the circular process allowed me to experience “ah ha” moments in my learning. I found excitement in accomplishment, discovery and seeing what my group members would bring to the group next. All in all this gives me great hope for what my students will experience through this being incorporated in my classroom.

August 4th,2006

Today is the final day of the Summer Algebra project workshop . All of our events again reinforce and support the importance of discussion, investigation, reflection, and written commentary. We used a new event to start the process : event; reflection; abstract; and application.

After the event and discussion: which should be rich and full; next during the "reflection" students use their own vocabulary and verbal style to reflect and describe the event: this is "people talk"; after extensive "people talk" ; Iconic representation is explored, this allows the student to create their own visual representation of the event. Next, during the "abstract" representations are also created for "feature talk": the language that belongs to student,teacher, and content specialist or experts in the area.

This week the event that took us through the Algebra project process was a math/science event/investigation facilitated by Garry Beneson, professor of engeneering at CCNY. During this event we examined mechanisms that are common, easily accessible, and familiar to all students. During the event we had to look carefully/attend to observing the different items. What attracted us individually to any particular mechanism.

After we selected individual objects, discussed and reflected on them, as a group we selected one item to discuss how it worked or how it was mechanical. We then had to find a way to make the object easier or harder to do its job. As we did that it was necessary to focus on each of the working parts of the mechanism. That process was scientific. My team had to describe what made it easier or harder to use scissors to cut materials. In the beginning we were overly detailed as to what was the moving part of scissor that actuallly cut the item.

The vocabulary that came out of our discuusion included vertex, connecting dot, pivot, etc.
Finally we used our icons and from the icons into mathematical representations effort, work, and the mechanical advantage.

July 7th, 2006

During the session today it was revealed to me how important a visual representation is to making an abstract algebraic representation ie. algebra sentence. **** Always following the sign is the benchmark. y1 - y0 = +4: compared to this benchmark the relationship is +4. Next , Where is Liz compared to Myra?


Quote from Dr. Moses "symbols are encoded as algebraic or arithmetic representations".

The main focus of the workshop for 3rd and 6th grade teachers so far has been to examine a curricula process to investigate the transition, in elementary mathematics, from arithmetical to algebraic ways of thinking.

July 6th, 2006

Today in the Algebra project we were reminded that the height chart when moved does not capture or retain height location or position. What is captured is relationships, difference, distance, and separation, as well as direction.

We discussed the importance of Habit building: Always revisit what you have written.
Designing a Worksheet.

Name/icon – direction -- height relationship/ distance/separation -- abstract-- Bicon.

People talk 7/6/06

Amy: The information is captured in the charting system… by displaying the relation of height to benchmark.

Pauline: The system of charting documents and annotates an event or happening. It is a permanent record that can be reexamined again and again.

Carol: There is a collection of data of individual benchmark heights.

July 6, 2006

Today we worked looked through the charts created comparing the heights of group members. After reflecting upon our charts it was interesting to notice what relationships were captured in the individual charts. The people talk of my group analyzed what relationship we had as a benchmark to our other group members. Carol noticed that her height was below average in comparison with the group heights. This addition of the vocabulary of the word average created a lot of dialogue throughout the group. We asked ourselves what the word meant to us and how we interpreted what Carol said.

One thing that stuck with me today was when Dr. Moses addressed the issue in schools with students and their writing. As teachers it is important for us to create a love of writing for our students. Dr. Moses said that through our project students need to internalize that, “what you write it worthy of your attention.” I found that to be a very interesting statement that each student internalizing that quote is a goal for me to achieve with my class next year.

I joined the group just as they were finishing their discussion of the height measurements' chart and beginning to design icons to represent different forms of information, i.e. person's name, taller than, shorter than. Dr. Moses led the group through the process of translating "ordinary discourse (Bob is taller than Arleen.)" into "regimented language (The height of Bob is greater than the height of Arleen)" then into the"abstract symbolic representation" of the sentence. This was a big ah-ha moment for me as I realized the importance of the "regimented language" step as a bridge between "ordinary discourse" and "abstract symbolic representation" which can support the development of students' conceptual understandings.

Elementary teachers workshop Day 1.

We began the workshop with a common exercise that engaged us as active learners.

Exercise 1. Using colored markers and large post-it notes, mark the heights of all workshops participants and list their names next to the marks.

Exercise 2. Write a paragraph or two about the exercise.

• REFLECTION: Think of the exercise as an event and reflect on what you did. Describe what you did.
• DISCUSSION: Discuss among yourselves, and talk about what you did.
• WRITING: Write a pararaph about what you did.
• READING: Read it aloud to each other.

We used 1" squared large Post-It wall chart paper, and attached to the wall. A line was drawn down the middle of the page. One person served as the "measurer", and with coloured markers, marked off the height of each person in the room.

I am in the middle of the group, and tied with Carol. Ron was the tallest, and Mary Lou the shortest. The difference between the tallest and the shortest person in the room was 12 boxes, which is 12 inches. Mary Lou is 1 foot shorter than Ron.

Making Connections: Explicitly connect children's everyday language to the more formal rigid language that is the literal translation of the symbolic algebraic/math language .

Exericise: Have 2 people of different heights stand and have audience/class observe and discuss the differences. Create/Write 2 sentences about these 2 people that explain/make explicit these differences.

(a) Ron is taller than Dawn

(b) Dawn is shorter than Ron.

What is the feature that is being talked/written about? Answer = Height

Write 2 more sentences but include the word height at the beginning of each sentence.

(c) The height of Ron is greater than the height of Dawn

(d) The height of Dawn is less than the height of Ron

What words are included in the second pair of sentences that are found in "mathematical language"

Answer = "Is greater than" and "Is less than".

Do those words "Is greater than" and "Is less than" convey information about the feature that is being compared/discussed? "Is greater than" and "Is less than" could be used to describe the weights of these two people, the length of their hair, the size of their feet...anything.

Notice that in this formal mathemtatical way of expressing oneself, there is an explicit decomposing of "is taller than" to include specfic language to express the feature "height" as well as the subjects of the sentence about whom we are interested in.

Problem solving in mathematics requires that we extract all of the pertinent information to be able to solve the problem. This requires a level of specificity and an attention to detail that is not normally present in every day language. He's taller. Spoken language has a brevity to it that is annoying when not used.

Mathematical language uses the extended version of the language. "He's taller" is enough in spoken language because there are other pieces of information that are being transmitted through other senses. Maybe an inflexion in the voice, a nod in the diretion of the person being spoken about etc. In mat, "he's taller" does not convey all of the information needed. We need to know that there are 2 people being compared, and that the feature or attribute that is being talked about is height, not weight, not age, not pigment etc, etc. Therefore we have a very formal stylised way of expressing the information which can then be translated into algebraic symbols.

(e) The height of Ron is greater than the height of Dawn

(f) The height of Dawn is less than the height of Ron

Who in the world walks around expressing themselves in this way? The same information in sentences (e) and (f) is transmitted in sentences (a) and (b). There is no need in every day language between people, to be that rigid or formal in trying to convey the information about the differences in height between Dawn and Ron.

But this rigid formalised language is the implicit albeit latent link between spoken everyday language and the symbolic language of algebra and mathematics.

Moving on from the rigid formalised sentence structure of sentences (e) and (f), we can create symbolic representations ( short hand symbols) for each of the components of the sentence.

In the classroom, children have the opportunity for artisitic expression creating their own iconic symbols to represent themseleves, their classmates, the features being discussed and the relationships being expressed. I am using the fonts available online to create my symbols.

• Create a symbol for Dawn: Here is my symbol d
• Create a symbol for Ron: Here is my symbol m

Translating the everyday language sentences (a) and (b) to symbolic language we now have

(a1) m is taller than d and (b1) d is shorter than m

We need symbols for "is taller than" and " is shorter than". This is different from the conventional mathematical symbols that we could use to represent " is greater than" and " is less than". The phrases "is taller than" and " is shorter than" contain information that is not included in the phrases "is greater than" and " is less than".

So my symbol for "is taller than" is 5 and " is shorter than" is 6

Translated to a symbols, Sentence (a) now becomes m 5 d and sentence (b) d 6 m

However, the rigid formalised language of mathematics extracts the feature, in our case height, from the everyday common usage of "is taller than" and " is shorter than", and places the word or icon for height next to the object that it is describes. An adjective if you like, ....adjectives in algebra... there's an insight.... so an adjective must describe a noun... height must be attached to either Ron or Dawn .

To translate the more formal sentences (c) and (d) into symbolic language, we need a symbol to represent height.

Create a symbol for Height : Here's my symbol .

Translating sentences (e) and (f) into my symbolic language we get:

(e1) . m is greater than . d

(f1) . d is less than . m

I can now replace the words " is greater than" and " is less than" and use the conventional math notation for " greater than" > and "less than" < .

(e2) . m > . d

(f2) . d < . m

The curricular process is a marriage of two research traditions:

• The first, Experiential Learning, is well known and is based in the work, among others, of Piaget, Dewey and Lewin. This work took off in the early grades (K-2) in the beginning of the 20th century. From this work we get a model for Experiential Learning that identifies four critical steps in a spiraling process, shown as four points on a circle.
  
  

At noon an Event takes place.

At quarter past there is a process of reflection on the event.

At half past there is the abstract conceptualization of the reflection.

At quarter-to there is an application of the conceptualization


Experiential learning in later grades has been used to drive subjects other than math; in these applications math has been an add-on using data collected and constructing some statistical analysis of it.

• I came across the second tradition in graduate school at Harvard in 1956 -1958 in the writings of W.V.O. Quine, who worked in Mathematical Logic as well as Philosophy of Math. Quine made the point that Elementary Logic, Set-Theory and Arithmetic, got off the ground by what he called “The regimentation of Ordinary Discourse”. You take the ordinary discourse that people use and, so to speak, straight jacket it. What you end up with is a conceptual language that nobody speaks, but that undergirds the symbolic representations of math and science. We inserted the transitions in language from “ordinary discourse” to “regimented language” to “abstract symbolic representations” between quarter past and half-past in the circle model of experiential learning.
  

Quine is part of a tradition of research mathematicians who argue for a “conceptual” and/or “intuitive” and/or “experiential” and/or “observational” input into mathematical truth. For example, Gauss, Kroneker, Frege, Klein, Zermelo, Weil, Goedel, etc. In school mathematics we have the question of what evidence we offer students and/or ask of students for the mathematical statements we ask them to accept: e.g. a negative times a negative equals a positive.

Describing the event.

As a group we randomly lined up and had our individual heights documented by placing our names next to the line that indicated our height. It was only after we looked at the chart that I made the mental connection of who shorter or taller.
As we talked among ourselves about the chart and what data was displayed, it was evident that the data had two outliers, shortest and tallest. The rest of the data was more evenly divided with the largest group being closely grouped by height.


The middle group represented a cluster of similar heights. We were told to think of pictorial representations of our heights. We were also instructed to think about our activities as an event that was discussed orally, written and read about. We were also instructed to make sure that there was and individual and group reflection.



*** The importance of talking, sharing; reporting out as a public person in the classroom not as a private activity.


Important vocabulary: symbols, iconic, generic.
Sentences used this session.
Bob is taller than Arleen.
Arleen is shorter than Bob.
Bob is shorter than Ron.
Ron is taller than Arleen..
Arleen is taller than Pauline.



July 6th, 2006

Today in the Algebra project we were reminded that the height chart when moved does not capture or retain height location or position. What is captured is relationships, difference, distance, and separation, as well as direction.

We discussed the importance of Habit building: Always revisit what you have written.
Designing a Worksheet.

Name/icon – direction -- height relationship/ distance/separation -- abstract-- Bicon.

People talk 7/6/06

Amy: The information is captured in the charting system… by displaying the relation of height to benchmark.

Pauline: The system of charting documents and annotates an event or happening. It is a permanent record that can be reexamined again and again.

Carol: There is a collection of data of individual benchmark heights.

ALGEBRA PROJECT:














We began by randomly lining up and marking our heights on chart paper. Each of us wrote our names by our mark to document our heights. We then engaged in conversation about the process of marking our heights, and what we noticed about the data. We first discussed the difference between actual height and perceived height. Collectively, we were surprised at the cluster of women around the same height. Arlene and I shared that the average height for women is 5”4. While we didn’t quantify our heights, our average height (where most of the data clustered), is taller than the national average. We knew this because those of us who fell in the cluster are around 5”6-5”7. Walking into the room, I would never have noticed that I was amongst a group of people who are considered “tall”.
Arlene noticed that the data would create a bell curve if plotted. The outliers of the data were the tallest person, and the shortest.
From there, we brainstormed ways to artistically represent the process of marking our heights. We decided on an image, and with the help of our artist Dawn, we sketched it on chart paper. The image showed the silhouettes of several people stacked one on top of the other, sort of like Russian dolls; the smallest in the middle, the tallest on the outside.
We also discussed and created symbols to represent ourselves, a generic person, and the concept “height”. More challenging was trying to think of a concrete symbol for the concept “is taller than”. We extracted a portion of our “height” symbol (up arrows with a bar above them) and changed it to represent “is taller than”.

Writing at beginning

I took off my shoes and stood there.

The lady put a stick on my head and put a mark on the paper. At the end, after everyone’s mark is taken, I was second to the shortest.

Discussion

12 --- Ron and Mary’s differences
4 --- Mary is 4 boxes shorter
8 --- Ron is 8 boxes shorter

Summary

We found that in our group we have both the tallest and shortest in the room. Ron is the tallest person in class and Mary is the shortest person in the class. Mary is 12 boxes shorter than Ron. I am 4 boxes taller than Mary. This makes me 8 boxes shorter than Ron. Dawn’s height is somewhere between Ron and I.