If you've had a chance to read the last two issues of this newsletter, you'll know that Pat Hess has been forwarding articles about her visits to Exxon project sites in Murray, Utah, and in Irving, Bellaire, and Fort Bend, Texas. This time she reports on her visit to Columbia, Missouri. For more on the Columbia project, see Linda Coutts's article in last month's Intersection.Ed.
Paul Trafton, project director from the University of Northern Iowa, has been conducting a study group in Columbia, Missouri. Linda Coutts, mathematics coordinator for the Columbia public schools, has provided two sessions for interested teachers, one during the day and the other after school.
Paul began his session with a reading and discussion concerning the idea of understanding. He used some quotes from Making Sense: Teaching and Learning Mathematics with Understanding by James Hiebert et al. (Heinemann 1997; $22.50) to start the conversation.
The teachers had had three and a half months to examine children's work after Paul's last visit. Here are the teachers' responses to the question: "How have your children changed this time?"
"They're more flexible in their thinking."
"They can transfer their thinking from one situation to another."
"They use more strategies, and are comfortable with making mistakes."
"They feel good about investigating different strategies."
"They can begin to look for more efficient ways of solving problems."
"The more they talk, the more they seem to challenge one another."
"The math becomes theirs. They have a joy of learning. They don't work just to please the teacher."
"Sometimes they challenge the teacher."
The enthusiasm of these teachers reflecting on their students' work was infectious. Ideas flew around the room.
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One seemingly simple, but interesting endeavor was to find the number of holes on a Chinese checkerboard. The instructions say not to count by ones. These first- and second-graders had been concentrating on the big idea of grouping, so the expectation of the teachers was that they would find patterns of ten, of five, etc. Most of the children did respond in this way, but some took the instructions literally and would not find a pattern of five's because they thought they weren't supposed to count by one to find the five's. These students were carefully counting by two's or three's.
One child could not make her teacher understand her thinking, and when the teacher said she would ask Dr. Trafton that afternoon, the student said, "Yeah, he'll understand."
Another child said, "Hey, this is symmetrical, so double it. Use doubles for all except one line."
Reproduced here are a few of the ways children solved the problem.
Many thanks to Pat Baggett, professor in the Department of Math Sciences at New Mexico State University, and to Karin Matray, the project director in Las Cruces, NM and director of the Las Cruces Teachers' Center, for the two contributions below. Since 1995, Pat and Karin have sustained meaningful collaborative endeavors between the local schools and New Mexico State University. Ed.
We offered our first university partnership course at New Mexico State University in fall 1995. Karin Matray located about 15 Exxon mathematics specialists (K–8 teachers) who enrolled in a late-afternoon graduate math class that met jointly with Math 111: Fundamentals of Elementary Mathematics I, a course required of undergraduate elementary-education majors. We received a private donation to pay tuition of the teachers. Acting as mentors, the teachers allowed the undergraduates to observe in their classrooms, co-teach with them, and eventually teach alone under their supervision.
We are now in our eighth semester of such courses. For five semesters we offered only one joint course each semester, either the course mentioned above, or Math 112: Fundamentals of Elementary Mathematics II. But since spring 1998, we have offered two courses per semester. Our newest course, sponsored by the Exxon Education Foundation, is called Algebra and Geometry for K–8 Teachers. It is now in its third semester.
Over the course of eight semesters, more than 400 people have been enrolled in the courses—about 250 undergraduates and 150 teachers. The Las Cruces Teachers' Center advertises the courses in the public schools and helps with recruitment. Several teachers have taken all three courses (earning nine hours of graduate mathematics credit), and they are asking for more. Once or twice in the summer, we hold get-together's at the Teachers' Center for teachers who have completed the courses.
In cooperation with Professor Rick Scott of NMSU's College of Education, we are developing a Master of Arts in Teaching with a specialty in elementary mathematics. This will require 18 hours of graduate mathematics. In the fall, we hope to offer a new partnership course, Elementary Mathematics with Technology.
The partnership courses are not typical lecture courses. The units we go over in class are almost all hands-on, with challenging mathematics content. Four-operation and non-graphing scientific calculators are often used. We work in groups and create lots of artifacts. Many units come from two books (by myself and my colleague Andrzej Ehrenfeucht). (See the paragraph at the end of this article for information about the books. Ed.) Other units have been and continue to be developed. You can see a small sample of the units on our budding web page, being built by an NMSU graduate student, Vanessa Galarza. Find it at http://charon.nmsu.edu/~vgalarza/MATH and click on the golden spiral.
One feature of each university class is classroom reports. As I mentioned above, undergraduates go into classrooms of teachers enrolled in the class and try with children the units we have gone over in the university class. Afterwards they report what happened back in the university class. These reports often lead to suggestions of what teachers can do to improve the lessons.
Also we ask the children to write what they remember about the lessons so that we can evaluate what they learn from a specific unit. We analyze what they write with no names attached and a great respect for confidentiality. These "recalls" are wonderful, and sometimes very funny. For example, after one unit titled "Chambered Nautilus" (made with right triangles), a fourth-grader wrote, "I learned that triangles are not boring!"
Another feature of the class is that students keep journals. If they try a unit in a classroom, they write about it in their journals. Here is an "old chestnut" math problem: "I looked in a park, and there were some children and dogs. I saw 7 heads and 20 legs. How many children and how many dogs were in the park?"
Here is what a kindergarten teacher wrote in her journal after trying this problem in her class:
"The children used 7 circles made out of construction paper for heads, and 20 brown rectangles for legs. So that no parts would be lost, a large piece of green construction paper was used as background to glue on the circles and rectangles.
"I told the children to imagine that in the park there were some children playing with dogs. If we counted all the heads in the park there would be 7. So the children glued the 7 heads onto the 'grass' (green construction paper). Then I told the children that if we counted all the legs there would be 20. I asked the children how many legs people have. They answered, '2.' Then I asked them how many legs dogs have and they answered '4.'
"I told the children they must use all of the legs. The children were allowed to work with each other. Kinders like to have their own work, so they each had a paper, but did a lot of talking. Many of the kinders were able to complete the problem, but others were distracted. Some of the children took the initiative to count and write totals of people and dogs on their papers.
"How the lesson might be improved: The children enjoyed figuring out this activity. Once they did, they were really proud of themselves. I would like to continue problem-solving projects with the kinders like this. There is a lot of math involved in this activity for kinders.
"My reactions to it: This project worked well. I used construction paper rather than pennies and peanuts (one way that we had solved the problem in the university class) because the kinders would become too distracted with eating the peanuts and counting the pennies. It worked very well with the shapes."
We will hold a Mathematics Education Institute at NMSU from April 10–13, 1999. Workshops with our materials will be offered, and attendees will come to our university classes and visit school classrooms when units are being tried with children. We still have room for a few more people (with very modest funding available). If any of you are interested, please contact Pat Baggett by phone at (505) 646-2039, or by e-mail at baggett@nmsu.edu.
Here's information about the books referenced above. Ed.
Baggett, P. & Ehrenfeucht, A. (1995). Breaking Away from the Math Book: Creative Projects for Grades K–6.
Baggett, P. & Ehrenfeucht, A. (1998). Breaking Away from the Math Book II: More Creative Projects for Grades K–8.
Both are published by Technomic Publishing Co., 851 New Holland Avenue, Box 3535, Lancaster, PA 17604, 1-800-233-9936 (8 AM–5 PM Eastern time).
The year 2000–01 will be a math adoption year for New Mexico. We have been struggling here (as folks are elsewhere, I am sure) with criticism of "problem-solving" and activity-based math programs. The public has questions: "Where is the math? Why aren't you teaching computation skills?" Principals and administrators are concerned about standardized test scores. As a result, teachers are feeling pressured to find instructional resources that are geared for improving scores.
Cathy Kinzer (one of our mathematics specialists) helped me design an afternoon that would give us a better picture of our purpose and our mission. Below is an account of what we did.
We convened a meeting of our mathematics specialists and a group of teachers that have been trained in Investigations. We also invited our new Director of Instruction. Forty-five people attended.
We grouped teachers so that every group had at least two "veteran" specialists. Then we asked the groups to compare and contrast issues (one per group) as to "Then and Now"—then being 1988–89 when our specialist group started, and now being 1998–99. Issues were: Pedagogy and Teacher Beliefs, Student Achievement Data, Teacher Support, Instructional Materials, Tools We Use, Accountability, and What We Know About How Children Learn. The small groups made huge charts with their comparisons and reported back to whole group. It was so cool! We had such a rich conversation!
After break, I gave a brief summary of the new NCTM draft Standards. We provided printed and bound copies for every participant (with the blessing of NCTM!). Everyone was delighted at the prospect of giving NCTM feedback. We divided into groups of three (self-choice) and did a jigsaw of the Pre-K–2 chapter. We asked the people in each group to take a section (that we had identified earlier), read it, discuss it, and identify three ideas that they wanted to share with the whole group.
It went very well. We left with a better understanding of where we have been and where we need to go, and all participants have a summary of one relevant piece of the Standards draft so that they can make educated choices.
We used my favorite evaluation form: 3 things you learned; 2 things you will use; 1 question you still have. Some of the evaluation responses were:
"We have come a long way! More to go!"
"There have been great changes in teaching math within the last ten years."
"As things in education change, they stay the same."
"The new Standards seem pretty user-friendly—and 'they' want input. Yay!"
"[I learned...] what colleagues are up to, how their year is going, what's happening in their buildings."
"Why adopt more books to use when people don't use them? Why not go from school to school with math workshops with the money?"
"When do we get to meet again?"
If you become a subscriber to the newly revived EXXONTNT listserv, you'll have an opportunity to participate in some rich conversations. Recent subjects have included: the value of sharing student work, mathematics textbook adoption, February's Kappan articles, teachers' trust and distrust of research, and class size.
It couldn't be easier to subscribe. Just send an e-mail message with these words in the body of the message: subscribe exxontnt. You can put those words in the subject line of the e-mail for your own purposes, but you will not be subscribed unless they appear in the body of the message. The account from which you send your message will be subscribed to the list. Please address the e-mail to majordomo@math.byu.edu. After subscribing, you'll receive additional information.
As you recall, Anne Herndon and Holli Aflatouni are the caretakers for the listserv. So that all subscribers may know one another a little better, Anne has made the following suggestion: "As the listserv begins again, I think it would be helpful for people who join up to give just a little bit of background info to the group: how you are affiliated with Exxon, your whereabouts, what do you on a daily basis, and some of your areas of interest. In this way, we may be able to 'place' people as they post information. It also make conferences much more fun when you can actually put a name with a face."
Even if you have been or are currently a subscriber to other listservs, you'll find something different with EXXONTNT. While its intention is simply to facilitate communication among members of the Exxon family, our listserv means much more than that to most subscribers.
To give you an idea of the meaningful conversations that take place out in cyberspace, directly below are a few edited transcriptions of subjects posted on the listserv.
Holli Aflatouni wrote:
"When you saw this question, I am willing to bet that you quickly answered, 'Yes.' I always assumed it was common knowledge that a square was a rectangle with four equal sides, but I realized the error of making assumptions a couple of weeks ago.
"My class was doing an Investigations lesson on making rectangles out of color tiles and seeing which numbers 1–12 made only one rectangle. This was a homework assignment after the children had made some explorations in class using color tiles. I felt pretty confident that my children understood their assignment and we even had a big discussion about squares being very special rectangles because they had four equal sides. Well, as soon as the children started arriving home, the phone calls started from parents who couldn't believe that I told their children that a square was a rectangle. Most parents were not angry; they were just confused because they thought a square was a square and a rectangle a rectangle. In fact, at dinner before a PTA meeting that night, another teacher had turned to me and said, 'A square is not a rectangle—they are completely different! The figure has to have two short sides and two long sides to be a rectangle!'
"I was simply amazed that any adult could make such a fundamentally wrong statement about an easy definition. When I argued with the other teacher at dinner, she told me that, 'She doesn't know about this new math stuff, but when she was in school they were two different things.' I was speechless, and that doesn't happen to me often, but a friend of mine defended me and said, 'The meaning of a square and rectangle is nothing new. It hasn't changed over the past 30 years since you were in school.'
"This hot debate continued the next day. I got notes from parents who argued with their children, and I am happy to say that most children stood their ground on a square being a rectangle! I went up to many teachers and various parents that I happened to see that day and asked them if they thought a square was a rectangle. First of all, math-phobic people are so apparent when you put them on the spot. They get this panicked look in their eyes. I was surprised to find that it was about 60–40 when I asked the question: 60% thought a square was a rectangle and 40% did not.
"At lunch in the teacher's lounge I brought it up again and you couldn't believe the loud discussion that took place in there! I truly believe that I have educated more adults than children on this subject now. I did make a big production in my math classes and looked up the definitions in the dictionary at my students' request so that we could clarify the subject. I had one parent tell me that it was the most meaningful math homework that they had ever done. It was certainly a learning experience for me.
"Does anyone else want to ask around at their school and among their parents and see what happens?"
Holli Aflatouni reported:
"Today was an exciting day because it was our Valentine's Party! This makes second-graders more bonkers than Christmas. I wanted to do something fun yet meaningful for all three of my math classes. I decided we would decorate Valentine boxes. So, I had all the children bring boxes, and I cut butcher paper for each child and just said, 'Cover and decorate your box.'
"It was very interesting. Some children were very haphazard, and just stuck randomly-cut out pieces of paper to the box until it was covered. Others who had some experience wrapping presents were very methodical and had smoothly-covered boxes that looked better than mine. Other children made 'nets' to cover their boxes. They measured, cut and folded until their boxes were neatly covered. One thing that struck me was the ability of some children to have a plan and carry it out.
"I was happy to see that several students who struggle with almost everything mathematically were really good at this. It was almost a no-brainer for them. And some students who always seem to have an innate mathematical sense about most things could not for the life of them get the paper to fit. One first-grade teacher who walked by and saw her son struggling with his box congratulated him on how good it looked, and came to me and said, 'Oh my gosh, his kindergarten teacher used to tell me he needed to work on spatial relations and I never realized why until now!'
"It was a wonderful experience for all the students because they had a real sense of pride about what they made, and I learned a lot from watching them. It also kept three classes of hyper eight-year olds busy for 45 minutes! It's definitely an activity I'll repeat."
Barbara Flynn responded:
"Your description shows how engaging the experience was. Two days per six weeks in art class is certainly not enough to satisfy students' needs to 'cut-and-paste.'
"Would you want to consider a follow-up to this activity that I have seen presented as a kindergarten activity? You can use any kind of boxes you like—cereal boxes, oatmeal containers, etc. Students cut them so that they will lie flat and can see the shape that was used to construct the container. Or they can do this outside of class to have other students 'guess' what shape the boxes make when folded into three dimensions.
"It seems to me that children do not have enough of these experiences to enable them to visualize 3-D shapes. One of the reasons that I like the Investigations curriculum is the richness of projects in geometry that it offers. Students' lack of experience is compounded when they get to high school geometry. All students must now take and pass geometry in order to graduate. Many high schools are struggling to find ways to help them understand."
Anne Herndon replied:
"In response to your Valentine boxes activity, Holli, I wonder with older children if you could give them the challenge to figure out how much paper they would need before they are given the paper?
"An extension that Bill Voss [the chief curator at the Fort Worth Museum of Science and History] thought of at 3 AM, and that we tried with teachers, was to give them a piece of paper and ask them to create an envelope with it. He threw envelopes out on the table, and had the teachers look at them after a few frustrating minutes to help us visualize how they were made. After we worked on the problem and created really ugly envelopes, we 'took apart' the real envelopes to look at their original shape. It was an interesting experiment with something we use every day."
Congratulations to Casilda Pardo, a Mathematics Resource Teacher (K–5) in Albuquerque, NM, who has accepted the invitation of Jane Swafford of the National Research Council to serve on the Committee on Mathematics Learning. At the current time, Casilda provides professional development for those at her own campus, Valle Vista Elementary School, and for teachers in other local schools as well.
The task of those appointed to the committee will be to conduct a study that uses research and scholarship as a basis for recommendations to improve student knowledge and understanding of mathematics in grades pre-K–8. The report of the committee, expected late this year, will be subjected to the NRC review process, which is exhaustive. The goals of the study are to:
One of the few teachers serving among individuals from various universities across the country, Casilda looks forward to working with the other committee members and representing the teacher perspective. Her hope is that the study will, indeed, make a difference in improving children's understanding and learning of mathematics.
Best wishes, Casilda! We look forward to hearing about your findings. Ed.
Intersection editor Jean Ehnebuske has been appointed by NCTM President Glenda Lappan to serve on one of three NCTM committees that have been established as part of the Public as Partners Group of the NSF/DOE Public Engagement and Understanding in Mathematics Initiative.
The overarching goal of the initiative is to help parents and members of the wider community understand and support what is important for students to know in mathematics by engaging them in conversations around a collection of interesting middle-school level "challenges." The idea is that these challenges will provide educators with opportunities to actually "talk to the public" about mathematics. Gail Burrill is serving as the Principal Investigator on the grant.
Three committees have been established by NCTM: a content advisory group, a writing group, and a "families as partners" group. Jean will be serving on the latter. The committee will meet for the first time in mid-March.
CESAME Support Site
The Center for the Enhancement of Science and Mathematics Education (CESAME) at Northeastern University is pleased to announce the launch of the "CESAME Support Site for Investigations in Number, Data, and Space". This web site is designed to support teachers implementing "Investigations," a K–5 mathematics curriculum developed by TERC in Cambridge, MA.
It offers a growing number of resources as well as discussion areas moderated by experienced "Investigations" teachers. CESAME is collaborating with the Math Forum and the Regional Lab at Brown University in this endeavor. The address for the web site is: http://www.lab.brown.edu/investigations. For more information, contact Peg Bondorew by e-mail at mbondore@lynx.neu.edu..
Who's Counting?
Check out http://abcnews.go.com/sections/science/WhosCounting/paulos990201.html. John Allen Paulos, a professor of mathematics at Temple University and the author of the recently released Once Upon a Number, writes an engaging column that appears there only on the first day of each month. His current subject is "A Tale of Two Numbers: From Y2K to PY2K."
Learning First Alliance
Pat Hess forwarded information she received about a report available at http://www.learningfirst.org. The Learning First Alliance, a consortium of twelve U.S. education associations, has produced on-line an action plan for improving the math skills of all U.S. students. "Every Child Mathematically Proficient: An Action Plan" summarizes the need for change in K–12 math instruction and provides recommendations for reforming math programs in key areas, including curriculum, professional development, parent involvement and tapping research-based reforms.
Call for NCTM Board Nominations
To find out if you or one of your colleagues qualify to serve on NCTM's Board of Directors, view the criteria, application procedures, and a nomination form at http://www.nctm.org. As NCTM states, "It is important that NCTM continue to be guided by people having leadership experience, knowledge of NCTM's activities and mission, and vision for the future." Clearly, those people are on the distribution list for this newsletter! Nomination forms are due by March 1.
DMI Summer Sessions
This summer will mark the second year of summer programs focused on professional development for mathematics educators. Two summer institutes, Developing Mathematical Ideas (DMI) Leadership Institutes I and II, will be held at Mount Holyoke College.
DMI L I is designed for staff developers, teacher educators, and teacher-leaders who are unfamiliar with the DMI curriculum (a program which addresses the mathematics of grades K–6). In this two week session, you will have the opportunity to experience DMI as a participant, meet with other teacher-educators to discuss issues in providing high-quality professional development in mathematics, and discuss the way you and your team can use DMI to further the mathematical agenda at your site. The session will meet from Sunday, July 11 to Friday, July 23. The $1500 cost includes room, board, and four graduate credits in mathematics education.
DMI L II is a one-week institute designed for those educators who are familiar with DMI. Educators who attended last summer's program are encouraged to return to this second summer of training. (If you conducted DMI seminars this year, you may want to expand your team by inviting some of your participants to attend the follow-up program with you.) In the one-week program, you will have the opportunity to discuss issues of facilitation, recruitment, and team-planning. All activities are designed for teams to work on developing structures and supports that will allow DMI to be used as a support for the mathematical reform agenda at their site. The one-week program will meet from July 17 to July 23 and will cost $1100.
The overlapping dates will allow for teams to include participants at both institutes. All participants will have access to an electronic network during the academic year that will enable the community that develops in the summer to continue. For further information or to request an application form, please e-mail smt-dmiinfo@ mhc.mtholyoke.edu or call (413) 538-2063. The deadline for applications is April 16.
Synergy Learning Summer Institutes
Synergy Learning is a not-for-profit organization whose mission is to engage teachers and learners in science/math/technology explorations which encourage them to become increasingly independent thinkers who are able to create positive change.
This summer two institutes are being offered. The first, "Gathering and Using Simple Data," for teachers of grades 2–8, is scheduled from June 28–July 2 in Brattleboro, VT. The second, "Science by Design," is offered for teachers of grades 2–6 in the Portland, ME area in early July.
The tuition for each session is $450, and includes a registration fee of $50. Enrollment in each session is limited to 25. Teams of two or more per school receive a 10% discount. For detailed information about graduate credit and descriptions of each institute, please see http://www.synergylearning.org.
or at the bottom, or in the middle of any page of this newsletter—for your article! If you've never contributed something for Intersection, or haven't contributed for a long time, why not consider sharing with your colleagues your thoughts about your project or a workshop or a meaningful journal article?
Please send articles for next month by Monday, March 15 to Jean Ehnebuske, 105 Hideaway Cove, Georgetown, TX 78628; phone, (512) 869- 1580; fax, (512)869-8477; e-mail, jehne@ibm.net. Thanks!
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