ET-MITT is a middle school teacher enhancement project (Empowering Teachers-Mathematical Inquiry Through Technology. The focus of the project is on the use of technology to foster an inquiry approach to teaching and learning. We are currently developping two graduate level courses for middle school teachers: Dynamic Explorations in Middle School Geometry and Interactive Algebra Activities for the Middle School. The courses will be taught at the University of Massachusetts, Lesley College and EDCO in the Spring, summer and fall of next year.
I work on the geometry part of the project collecting materials and software. I recently did a software presentation at EDCO on software that we`ve been exploring. I thought the material might be useful to some middle school teachers on the Forum. Warning! It`s a little messy - essentially, it`s just my notes for the presentation. .......
Since we`re trying to foster an inquiry approach, we`re interested in software that can be used as a tool, that has a more general, open approach, rather than something that`s directive.
Dynamic Geometry Software:
One of the main tools we`re using throughout our courses is a piece of dynamic geometry software. I`m sure you`ve all seen examples of some of these, the Supposers being the precursors to the others. Currently there exist The Geometry Inventor, Geometer`s Sketchpad and Cabri.
We`ve focused on Geometry Inventor because it`s easy to learn to use it (which is an issue - how long does it take to become proficient in the use of a piece of software) and because there`s plenty of opportunity within the use of the software for kids to move from constructing specific figures to general examples and we think the process of moving from specific to general is a very important one in the middle grades and relates well to the process of understanding variables in algebra.
There`s a new version of Inventor, not yet on the market. I like it because it has pre-constructed geometric figures which brings more geometry language in, and because it has a fixed perimeter and fixed area function.
(Build a rectangle with fixed perimeter and measure lenghth of 1 side, Move it to show change, measure perimeter, move, no change, area, move , change. Graph Side vs. area. Move shape around.)
A feature we think is important in software is being able to move information from other pieces of software into analyze them. You can move a picture into Inventor and then superimpose the construction and measuring tools to analyze shapes, measure angles, lengths, area, etc.
We like Inventor for many things, however, we have bumped up against some limitations and are currently beginning to look more closely at Sketchpad and Cabri. We should soon be getting some information from teachers who have used these pieces of software with 9 and 10 year old kids.
Geometry in Context:
The NCTM Standards call for increased attention to connections between mathematics and the real world. Geometry is a nice way to make that connection and software can be part of that. We`re using a number of different /contextualized softwares and beginning to look for ways that we can analyze the mathematics in others and use them in our classrooms.
There are a number of bridgebuilding and related activities out there, for example, EDC`s Structures curriculum and Alan Hoffer`s exploration of what makes a geometric figure rigid in Geometry and Visualization. Triangles are an integral part of truss bridges. Symmetry is a major ingredient of stability. And there`s probably lots more - like what kinds of triangles make sturdy bridges, vector analysis, etc.
A new piece of Mac software now makes it possible to test one`s ideas about bridges. Bridgebuilder.
(Build bridge that doesn`t work (Geometry unstable). Build bridge that gives numerical results. Talk about kids focusing on building an /efficient bridge, collecting data, isolating variables, varying things systematically - a good mathematical process for kids to go through.)
We`re working on a scale drawing unit in which we will design a dwelling with a given area. With all students using the same area for different designs, we can begin to explore the relationship between area and perimeter. Then we`ll focus on 3D and spatial visualization - the Middle Grades Mathematics Project has a nice unit on looking at buildings from different perspectives. For this unit we`ll use a simple CAD program called: Architecture - design your own home.
(Show floor plan) (Interiors demo showing 3D) (Landscape - don`t show - tell a bit about it - has the different views like interiors, set the age of the trees and see how they grow. I used it to model an orienteering course, which I then moved into Inventor so that I could create different courses of similar lengths.)
One of our activities is an exploration of combinatorics and symmetry in the construction of a quilt design. We use Geometry Inventor to explore geometric features like: similar triangles, congruence, perpendicularity, transformations. Then we move into a paint program to investigate the combinatorics of coloring in a certain number of pieces in a square and into a draw program to explore the relationship between lines of symmetry and numbers of different patterns you can create by putting squares together. Another nice piece of technology that has provided me with lots of information about the mathematics of quilts and other things is the internet. I`ve been involved in a user group called the Geometry Forum. Through it, I`ve gotten lots of valuable information about geomety curriculum and research, interesting geometry problems, connections to people who use geometry in real world applications. In fact, I`m involved in an internet group of math quilters. We`re all doing math blocks and trading them.
Paint and Draw programs (MacPaint, MacDraw, Aldus Superpaint)
I`ve come across a nice little piece of quilt software from the Mathematical Sciences Education Board. It goes with a booklet of assessment prototypes /Measuring UP and I`m thinking about it as a useful assessment tool for our quilt activity. Plus the fact that it`s really fun to play with and it`s CHEAP!
(Do one square and rotations)
(I am told you can FTP it off the network for free, but I haven`t figured out how to do that yet. I sent my $20 in and had them mail it to me!)
This past summer, as a result of my involvement on the Internet, I was invited to a Geometry Software Conference sponsored by the Geometry Center in Minnesota. A strong focus of the conference was the importance of 3D Geometry, a lament about the absence of 3D geometry in the curriculum and a call for greater focus on 3D geometry in grades K-college.
A number of people are working on 3D curriculum, including Mike Batista of Kent State in an elementary school project with TERC and Alan Hoffer. Alan demonstrated a piece of software he developed, 3D Images, and I was impressed by it. I had been looking at this piece of 3D software in the catalogue for a year, but hadn`t ordered it. My own 3D education has been pretty limited, so it took some time for me to overcome my 3D phobia and order this software. I`ve had it now for two weeks and I`m in love with it. The only thing is, I now want a color Mac so I can use a very important piece of this software (pull out 3D glasses).
Demo: Relationship between a square and face, extrude, revolution, Cone & double cone tools. Building regular polyhedra Build a sphere, show cross section, mark cross section, divide by cross section. Examine how a surface is created - goblet Face screen.
I`m not exactly sure how I`d want to use this software yet, but I`ve been thinking about it inrelation to some of the shadow activities that I`ve seen, for example a shadow activity that Dick Lesh from ETC presented at the Forum workshop this summer.
Something I think I`d like to have on this software is the ability to measure things (angles & distances & surface area) But I think I can get around this by copying and pasting a picture into one of the dynamic geometry software pieces, like Inventor and measuring there.
I`m relatively new to Logo, but I like it. I taught middle school before Logo became popular, went into business and have the last few years come back to a new and what I consider to be an exciting era in mathematics education.
There are a couple of pieces of Geometry/Logo software that are currently being developed. One is by Richard Lehrer. It links defining and constructing a figure with Logo, a really nice way for kids to work with the properties of geometric figures and relate them to the logo environment. This is not currently available on the market, but I hope a version of it does get finalized and the other is Geo/Logo, developed by Doug Clements . Doug Clements University of Buffalo- has previously developed (with Mike Batista) an elementary geometry curriculum based on Logo and is currently working on two new K-6 mathematics curriculum projects with people at TERC.
Geo-Logo will be marketed this coming year and combines many of the strengths of dynamic geometry software with Logo. It will come with many activities and is also going to be used with a curriculum (Connected Mathematics) which is being developed by the people who developed The Mouse and the Elephant - the Middle Grades Mathematics Project.
Short demo of main features. Can set and measure angle and distance, move turtle by hand and watch Logo language change, work on a grid, create a script, step the motions, etc.
I`m enclosing a list of suppliers for the software that you`ve seen and if you have any questions, you can call me at:
Geometry Inventor : Wings for Learning, 1600 Green Hills Rd., P.O. Box 660002, Scotts Valley, CA 95067-0002. (800) 321-7511 New version probably going to be distributed by Logal America - information coming soon.
Geometer`s Sketchpad - Key Curriculum Press, 2512 Martin Luther King, Jr. Way, Berkeley, CA 94704. (415) 548-2304
From Abracadata, P.O.Box 2440, Eugene, OR 97402. (800) 451-4871:
Architecture: Design Your Own Home Interiors Landscape Sprout Design Your Own Railroad
Paint and Draw programs (MacPaint, MacDraw, Aldus Superpaint)
Quilt Designer - Mathematical Sciences Education Board, National Academy of Sciences, 2101 Constition Ave NW, HA476, Washington, D.C. 20418.($20)
3D Images - Wm. K. Bradford & Co., Acton, MA. 1 (800) 421-2009
Geo/Logo - Doug Clements University of Buffalo Susan Einhorn, Director of Marketing, LSCI, 3300 Cote Vertu Road, Suite 201, Montreal, Quebec, Canada H4R 2B7. (800) 321-5646.
For additional information, contact Claire Groden at BBN Systems & Technologies Phone: (617) 873-4145 Email:email@example.com or Laurie Pattison-Gordon at BBN Systems & Technologies Phone: (617) 873-2695 Email:firstname.lastname@example.org
From usenet Sat Oct 02 12:30 PD 1993 Path: forum.swarthmore.edu!uunet!cdp!doversherhs From: Dover-Sherborn High School <email@example.com> Newsgroups: geometry.pre-college Date: 02 Oct 93 12:30 PDT Subject: DSHS Journ.-Filling in Gaps Message-ID: <firstname.lastname@example.org> X-APC-HostID: 1 Sender: Notesfile to Usenet Gateway <email@example.com> Lines: 215
October 1, 1993
Let's see if I can fill in some of the gaps.
At present I have access to a Mac and a piece of equipment (name unknown to me) that projects the computer screen onto the pull down screen in my room. I have used it so far for demonstrations of Sketchpad's capabilities, to help students understand ideas such as altitude, median, and perpendicular bisector, and to gradually acquaint the students with the tool box and menus.
The computer labs (I will be using two different labs), are still in the "set up" phase - including negotiating schedules, linking computers, loading programs, and dealing with the politics of schools. My most difficult hurdle still lies ahead - securing a computer room key for my key chain. For the past two years I have had to go to the computer room and ask a teacher in an adjacent room to open the door for me. More then once, I stood in the hall with 15-20 students (when the teacher was not available) while a student ran down to the office to have a janitor or administrator come to the room to open it. Security must be maintained.
Hopefully by next week, I will be taking my classes over. There are just enough computers so that the students may work in pairs.
Many students seem eager to begin using the computers, entering class each day asking if we will use computers that day.
Students are beginning to understand and become more comfortable with the requirements of the 'problem of the week' assignments (see written evaluation document). I have assigned four problems to date. The most resent was to answer the question: What is the relationship between the number of rays that are drawn from a single point and the number of angles formed?
Students had a chance to ask questions when the problem was assigned, were required to write a rough draft for homework, and then were given a chance to ask questions the next day in class.
Students are telling me that they have not had much experience solving problems - that they are threading in unfamiliar territory. In response to this, I xeroxed and distributed what I considered to be one of the best student papers from last week. We read through the paper together and identified important elements that contributed to its excellence (again referring to the written evaluation document).
I'm concerned that I am teaching students a "canned" approach to solving problems - in other words imposing a structure or lattice upon them. I would prefer a more open-ended, developmental process: asking students to pose questions or situations they wish to investigate, 'allowing' them to determine the course, set the sails, and navigate the wide open seas. I have taken this tact in the past and would have been satisfied with a lot of shipwrecks and scattered treasures. Instead, a vast majority of the students looked (and I promise not to stretch the metaphor beyond this point) lost at sea, unable to act on my instructions. There frustration levels were high - it was very difficult for them to "hang in" with me on the journey I was suggesting. I wound up providing them with the "canned" version in order to get them off the mark. Perhaps I have given up too soon and/or lacked the necessary experience. In any case, I have much to contemplate in this area. If this is at all coherence and of interest to anyone, I would of course welcome your ideas and questions. Perhaps someone could provide a "canned" road map for how not to provide students with a "canned" road map for solving problems!
EDC has written a draft of a paper concerning the Big Ideas in Geometry. In it the authors speak of mathematics as a super-set of ordinary language containing extra constructs and symbols for description. Michelle Manes - firstname.lastname@example.org - may be able to provide copies if anyone is interested. The document contains a lot of interesting ideas that I am finding useful as I attempt to provide frameworks and contexts for the work we do in the geometry classroom.
I used Serra's materials to help the students learn the first set of definitions. The widget exercise group and the inductive approach to learning the definitions. Serra provides three or four diagrams to represent a relationship, say triangles with unequal sides, three or four diagrams with un-similar characteristics, and asks students to write a "good" definition of a scalene triangle. His definition of "good" definition (precise, withstands counter-examples, and reversible) appears to be very useful for students.
After students completed two nights of writing definitions and two twenty minutes periods in their small groups discussing definitions, we spent 30 minutes as a class answering any remaining questions (I used Sketchpad here when I felt it would help.) The first evaluation on definitions, the scores ranged from 60% to 88%. I stressed that after only two weeks of class, everyone knew approximately 2 out of 3 geometric definitions (from a list of 75 or so). I reminded students to try and view the evaluation as a landmark on our journey toward becoming mathematician. (I didn't state it exactly like this!) If results were poor we would reflect on our habits and conduct an inventory of sorts; perhaps revise our strategies. (I think everyone gets the idea>)
Community (Collaborative Classroom):
We've agreed upon behavioral norms; we've agreed to meet periodically, or when necessary, for after school conferences. Students have shared their most powerful learning experiences and their math autobiographies. We have done two introductory exercises (which contained patterns which could be expressed as functions and arrived at through a collecting data and identifying relationships) and one group problem solving exercise called Zin Obelisk (I'll post this). Three students in each class have agreed to act as observers of our class and provide a brief written report - without using specific names for now. I was out of school Thursday and Friday of this week because my fifteen month old son, Brendan, was sick with the croup, so I'll have to report further on this at a later date.
Students were asked to write up the zin exercise as a 'problem of the week' and also to complete the Group Problem Solving document, including a paragraph or two, in order to provide information and data for reflecting on our process.
They seemed to have a lot of fun zin. I always have fun observing their interactions. Each year I'm amazed at what students are capable of doing when asked. And a couple of 'student facilitators' always seem to rise to the occasion. I'll say more about this later, but by the end of the first term - sometimes sooner, sometimes later - students facilitate class (to various degrees, at particular times and in certain circumstances).
Next week I plan to:
-spend 20 minutes or so of one class processing the zin exercise (beginning with a summary of their evaluations),
-depending on the results of the definitions evaluation (which by the way also included a couple of inductive exercises) do something different with definitions,
-begin constructions using straightedge and compass and Sketchpad,
-spend one class doing three exercises I use dealing with perceptions. - in one I present students which various arrangements of line segments and ask them to identify which are longer; of course they are arranged in such a way that the eye and mind interpret the relationships incorrectly in at least half the case for most students (and adults). - in another I present students with a sketch of a woman - some students see an old woman, some see a young woman. - in the third I simply draw a representation of a cube on the board (with all solid lines) and ask them to describe the cube they see - which face of the cube is closest to them or farthest away. When I ask them to look again for another cube 'within' the same lines, about half see it within seconds, followed by a chorus of 'wows' and 'neat'. The other half, say things like 'what?', 'where?' and 'I don't see it. It's fun - and of course several interesting conclusions can be draw from the exercises - which I ask the students to write about and share.
In trying to fill in the gaps, I realize that I have created many others (i.e. How am I planning to introduce constructions?) I do have other plans and strategies and would love to share them all, but time marches on and I'm marching to bed.
(I hope this is in some way interesting to someone besides me?!)
In the ancient city of Atlantis, a solid, rectangular obelisk, called a zin was built in honor of the goddess Tina. The structure took less than two weeks to complete.
The task of your team is to determine on which day of the week the obelisk was completed.
You will be given slips of paper containing information related to the task. You may share this information orally, but you may not show your cards to other team members.
*The basic measurement of time in Atlantis is a day. *An Atlantian day is divided into schlibs and ponks. *The length of the zin is 50 feet. *The height of the zin is 100 feet. *The width of the zin is 10 feet. *The zin is built of stone blocks. *Each block is 1 cubic foot. *Day 1 in Atlantian week is called Aquaday. *Day 2 in the Atlantian week is called Neptiminus. *Day 3 in the Atlantian week is called Sharkday. *Day 4 in the Atlantian week is called Mermaidday. *Day 5 in the Atlantian week is called Daydoldrum. *There are five days in an Atlantian week. *The working day has 9 schlibs. *Each worker takes rest periods during the working day totaling16 ponks. *There are 8 ponks in a schlib. *Workers each lay 150 blocks per schlib. *At any time when work is taking place there is a gang of 9 people on site. *One member of each gang has religious duties and does not lay blocks. *No work takes place on Daydoldrum. *The zin is made up of green blocks. *Green has special religious significance on Mermaidday. *Each gang includes two women. *Work starts at daybreak on Aquaday. *Only one gang is working on the construction of the zin. *There are eight gold scales in a gold fin. *Each block costs 2 gold fins.
From usenet Mon Oct 11 18:10 PD 1993 Path: forum.swarthmore.edu!uunet!cdp!doversherhs From: Dover-Sherborn High School <email@example.com> Newsgroups: geometry.pre-college Date: 11 Oct 93 18:10 PDT Subject: test - posting problem Message-ID: <firstname.lastname@example.org> X-APC-HostID: 1 Sender: Notesfile to Usenet Gateway <email@example.com> Lines: 8
This is part two of a two part test.
Apparently not all of my postings are actually making it to the Geometry Forum.
Since I post through two different avenues, I am testing each.
This is posted through my Peacenet account.
From usenet Tue Oct 12 15:29 PD 1993 Path: forum.swarthmore.edu!news.bu.edu!decwrl!decwrl!uunet!cdp!doversherhs From: Dover-Sherborn High School <firstname.lastname@example.org> Newsgroups: geometry.pre-college Date: 12 Oct 93 15:29 PDT Subject: test-wellesley Message-ID: <email@example.com> X-APC-HostID: 1 Sender: Notesfile to Usenet Gateway <firstname.lastname@example.org> Lines: 8
This is part one of a two part test.
Apparently not all of my postings are actually making it to the Geometry Forum.
Since I post through two different avenues, I am testing each.