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NCTM’s Winning Theme ParkIn a city noted for over ninety attractions, the 79th Annual Meeting of the National Council of Teachers of Mathematics — held in Orlando from April 4 – 7 — was the preferred tourist destination for approximately 16,500 mathematics educators from across North America and around the globe. The conference theme, "Math World: New Standards for the New Millennium," was the unifying concept that brought educators together to attend sessions, participate in workshops, hear keynote speakers, view new materials, network with colleagues, and sample Orlando’s hospitality.
After greeting an audience of two-thousand at Wednesday’s Opening Session, NCTM President Lee Stiff officially convened the Annual Meeting. Dr. Stiff recognized and presented all NCTM board members — outgoing, remaining, and newly-elected. He also introduced Johnny Lott, NCTM’s president-elect; Glenda Lappan, NCTM’s past-president; and John Thorpe, NCTM’s retiring executive director. Then Lee Stiff introduced Richard Priory, CEO of Duke Energy, who announced that his company would be joining NCTM in creating a web-based professional development program for college credit. The program, Reflections, is expected to be up and running by April 2001.
Acknowledging the publication of the Principles and Standards for School Mathematics — unveiled by Glenda Lappan at last year’s conference — as "one of our most significant accomplishments," Dr. Stiff noted that we still have a long way to go to see that each child receives a meaningful mathematics education. At this point, he announced the debut of the first four of thirty planned titles in the new Navigations series. These are the grade-specific support materials (similar to the Addenda series) developed to help teachers put into practice the individual standards and principles in PSSM. The first four books — which include CD-ROMs — focus on Algebra across the grade bands (pre-K–2; 3–5; 6–8; 9–12). (To order, call 1-800-235-7566 or visit NCTM's web site. Ed.)
There are hardly enough superlatives to describe this year’s meeting. To learn more, you can find session descriptions, selected handouts, web casts of major sessions, and news clips at NCTM's web site. Better yet, turn to the annual supplement for this issue, "A Collection of Reflections." Some readers who were fortunate enough to attend the conference have been generous enough to share their most meaningful experiences there. It is their words that best capture what it was all about.
If pictures speak a thousand words,
then the smiles on the faces in the photos scattered throughout
this issue say it all — those who attended ExxonMobil
Foundation’s traditional reception on Thursday night truly
enjoyed themselves!
There was an abundance of noshing and networking as guests greeted old friends and met new ones. Hosted in the Signature II Room of the Rosen Centre Hotel, this annual come-and-go event allowed guests a unique opportunity in the midst of the conference to get to visit with one another over drinks and a delicious and bountiful repast.
ExxonMobil Foundation’s Joe Gonzales welcomed guests with these words: "We’re proud of who you are and what you do. Thanks to you, parents and community members see children performing in ways that they didn’t think possible. We thank you all for your work, and we’re pleased that you’re here tonight."
Following Joe, Jean Moon — Advisor to the Foundation — conveyed her greetings: "It is such a pleasure to see all of you and to meet new people. With our strong network of 115 sites in 30 states and the Netherlands, you are part of something large and important. Stay the course. And please enjoy this evening."
It was an honor to have the following
individuals drop in for conversation and refreshments: Lee Stiff,
current NCTM president; Johnny Lott, president-elect; Shirley
Frye, president in 1989; John Thorpe, executive director; and
Cindy Chapman, new board member. Coaxed to the microphone to make
a few comments, Dr. Stiff said, "I’m delighted to be
here with you, and I’m pleased that the ExxonMobil
Foundation has brought you together to celebrate the work you
do."
Johnny Lott also thanked the Foundation and made informal remarks at the mike. He offered, "If there is ever anything that I can do to help you, please send me an e-mail."
After expressing her thanks to the supporters who helped elect her, Cindy Chapman thanked the Foundation, too. "If it weren’t for my experience in our ExxonMobil project, I don’t think I’d be on the board today."
Guests continued to enjoy one another’s company until the reception ended shortly after 8 PM.
Many thanks to Emily Dann of Rutgers University for this article about her work with teachers in the Colts Neck, NJ schools. Ed.
Last winter when it was requested that I conduct workshops on assessment with Colts Neck teachers, it made me think of the work on paper-and-pencil assessment done at the Freudenthal Institute in the Netherlands. The work at the Freudenthal has been crystallized by Maria van den Heuvel-Penbuizen in her book Assessment and Realistic Mathematics Education. The ideas put forth there add a distinct perspective to the search for better assessment of instruments.
The Colts Neck teachers involved were grades 1 – 5 teachers who are engaged in implementing the TERC Investigations curriculum and who take part in the ExxonMobil Foundation-funded project. The aim of teachers and administrators in the Colts Neck schools has been to provide richer and more open problems that engage students at many levels. The purpose of this has been to broaden student understanding of the inherent situational mathematical properties. Could we capture this in a testing situation? I decided to share with teachers some of the ideas put forward in van den Heuvel-Panbuizen’s book and subsequently work with these teachers to create tests that would help them assess student learning with the TERC Investigations materials being used in their classrooms.
The focus of the Freudenthal assessment work has been to move written assessment to be more active, "elastic" and rich. Suggestions from the book about accomplishing this include opening up problems to have more flexibility in selecting what problem or facet of a problem is worked as well as accepting a variety of forms in both answers and methods of solution. Included in this work are several simple but helpful techniques. Students can, for example, be asked which of several strategies they might use to solve a problem or produce some calculation. Some formats can allow for more than one answer or even more than one choice of problem ingredients. Students can be given options if they find one question too difficult or too easy.
Alternatively, given a set of constraints, students can be asked to create their own productions. Paired tasks could be created that allow students to work first on one easier task and then another where a solution to the second problem could, with insight, be more easily worked using results of the first. In all of this there can be sections on the page called "scratch paper" where students are expected to do their work enabling teachers to assess more components of the child’s conceptual understanding. In another facet of the Freudenthal assessment work, attention was given to considering how interview techniques could be used for paper-and-pencil tests. Considered were "safety net" or "second chance" questions that might be used or having a "standby sheet" (separate page) with extra questions for guiding student thinking.
After reviewing these ideas with the Colts Neck workshop participants, teachers were challenged to develop tests for their students that included more open tasks, possibly using these new ideas. Instructor creations ranged from somewhat usual to unique and imaginative. All are charming. They provide a more interesting activity for these young students and capture the flavor of the mathematics being studied in these classes form the Investigations curriculum.
Teachers met in small grade-level groups for these work sessions. One of these groups created a test around the theme of "Planning a Birthday Party." Questions they asked included:
"Invitations come in packages of 8. If you have 24 friends to invite to your party, how many packages do you need to buy?"
"If you hand out 14 of your invitations at school and mail the rest, how many stamps do you need to buy?"
"If each stamp costs 33 cents, how much money will you spend on stamps?"
"There will be a magician at the party and you will be his assistant. You will need to arrange 24 chairs in even rows for your friends to sit on during the show. Show 3 different ways that you could arrange the chairs."
"Which arrangement do you think would work best? Explain your choice."
Due to teacher enthusiasm, most of the tests turned out too long for the children, and had to be given in pieces. Many of the problems reflected ideas from the Investigations materials. For example, an array problem displayed 4 cakes cut in 4 different ways (4x6; 7x3; 6x5; 4x7), and asked children to decide which cake would be best for this particular party and why.
Other problems were motivated from alternative ideas discussed at the workshop. For instance, to check on their understanding of multiplication, children were asked to examine various pictures (e.g., arrangements of gingerbread men; groupings of shirts and pants; lines of identical egg cartons, etc.) and choose two that could be used to help teach a younger child about multiplication. Students were also asked to say how the picture could be useful.
Teachers have been using the test instruments and find that many of the problems give students more range to explain their thinking than tests that have been used in the past. Also, it has encouraged them to include testing as an extension of class work as well as a consolidating activity for the concepts being studied. I am satisfied that the teachers’ time was well used.
You may recall reading a transcript from the EXXONMOBILTNT listserv in the Nov/Dec 2000 issue (see Intersectionlive) in which Holli Hall related her second-graders’ thoughts about equality. Holli, a teacher in Arlington, TX — as well one of the listserv’s caretakers — has continued to share children’s discussions on equality on the listserv via e-mail. Below are some selections, slightly abridged due to space constraints. Thanks, Holli! Ed.
I find it interesting that this issue of equality came back all on its own. I’ll be interested to see where it goes from here.
Today Jimmy used the following problem as a way to make 98 (the number of days we have been in school):
(500 × 1) – 500 + 97 = 97 + 1 = 98
Most students agreed with him, but Hal and Tony did not. They both said they didn’t agree because you can’t have two equals signs in one problem. Other students said you could. I decided to ask again, "What does the equal sign mean?"
Ed said, "When there is an =, both sides have to be equal." Everyone seemed okay with that. Someone added that = is like a stop sign and it has to be at the end.
So I asked, "What about 5 + 5 = 10 and 10 = 5 + 5 — do those mean the same thing?" Everyone agreed that they do because either way each side is 10. Several students said that = is like a stop sign, but it can go anywhere in the problem.
Tom chimed in that he agreed with Jimmy’s problem and that he could explain why. He said that you do the first part and stop, you do the middle part and stop, and then you do the end.
So I asked what the first part equaled and everyone said 97. I asked what the middle part equaled and they said 98. Then they said that the last part equaled 98. That caused some dissention in the ranks.
Someone said, "Well, just stop at 97 then do the + 1 later."
I said, "Can you do it that way? I thought you said the = was the stop sign. Is the + a stop sign, too?"
Several people said if it had a +, you had to add the numbers togethery — you can’t just stop. At this point, Hal asked if we could stop because he was getting a headache. Then Winnie asked if we could use a simpler problema to see the same thing.
So I said, "Well, what about 5 + 5 = 6 + 4?"
Hal said, "No, add another = sign so there are two, like Jimmy did."
So I wrote 5 + 5 = 6 + 4 = 7 + 3. There was a general murmur of agreement. Jimmy said, "Well, my problem is just like that, why don’t you agree with me?" Then Jimmy said, "Oh, mine is not equal on all sides!"
Tony said, "If you change the first 97 to a 98, I will agree with you."
So we changed it, but some people said, "No, that doesn’t work, it needs to be a 97." Then someone asked if we had been working on this problem for 30 minutes yet and someone else said no, just 20 minutes. Then Hal mentioned his headache again. Jimmy said he wanted to leave the problem the way he had it originally so we put it back.
Tony said, "Okay, but I don’t agree anymore." We agreed to leave it alone and think about it some more.
Once again the subject of equality emerged in my classroom from a problem that Jimmy made up for our number of the day. Here it is:
138 – 0 + 0 + 1 – 1 = 137 + 1
Jimmy said, "Okay, does everyone agree with me?" Everyone agreed except Becky and Tom. I asked Tom why he didn’t agree.
He said, "Well, that first part equals 138 and Jimmy’s problem says it equals 137." A lot of students started saying that that wasn’t what the problem meant.
Sandy commented, "Tom, it doesn’t matter where the equal sign goes — each side is still the same in that problem."
"No, the first part is 138 and then the equals sign has 137 after it," Tom said.
Tony said, "Yeah, but it has 137 + 1, which is 138, just like the first part." Tom looked confused and didn’t know what to say. So I asked Becky why she disagreed. She said that at first she disagreed for the same reason as Tom, but that after Sandy had explained it, she remembered what the class had decided last time and agreed, too. I asked what the class had decided.
Becky said, "Ed said something about what the equals sign means but I can’t remember exactly what it was." Ed said he couldn’t exactly remember either. I remarked that I thought Ed said that whatever was on each side of the equals sign had to be balanced, or the same.
"Yeah, that was it," said Ed.
Becky said, "Right, so that problem is right."
Hal said that he understood why Tom disagreed, so I asked him to explain.
"Tom thinks the equals sign is like a stop sign and you only look at the 137 and not the + 1. I used to think that, too, so I see why he thinks that. But as long as both sides come out the same, it is equal. It doesn’t matter where the equal sign is." Murmurs of agreement all around except from Tom who looked frustrated.
I asked Tom if he wanted to ask or say anything else and he said no. Nobody else had anything to add so we just left the discussion there. There was a student from a local university in my room observing and at one point he went over to Tom to explain to him why he should agree with the problem. I stopped him and said, "No, don’t say anything. If Tom wants to disagree that is fine. We used to have a lot of people who disagreed with this kind of problem and that is okay." I took the college student aside later and explained the history of this discussion in my classroom. I told him that Tom has to decide on his own what he thinks. The college student remarked that he thought Tom would keep thinking it through. I said, "Tom’s understanding needs to come to him on his own terms, when he is ready. It’s okay for him to be frustrated and think about it."
Once again, the equality subject emerged. Here is the way the conversation went. Tony wrote:
141 + 1 – (16 × 1) – 0 = 141 + 1 + 7 – 7 – 1 = 141
Jimmy quipped, "He learned from the master."
I asked, "Does everyone agree?" Most say yes. Even Tom agreed, which surprised me. "Tom," I asked, "why do you agree with this one but you didn’t agree with Jimmy’s last week?"
"Because in this one the equals sign is at the end and in Jimmy’s it was in the middle," Tom said.
Sandy noted, "But Tom, it can be anywhere. And this one has an equal sign in the middle, too. It has two equals signs."
Tom said, "I think this one is okay because even though there is an equal sign in the middle, there is also one at the end."
"So," I said, "it’s okay to have one in the middle if there is one at the end?" Tom said it was.
Tony said, "All the parts still equal 141 and I agree with Sandy that the equals sign can be anywhere." I asked how many people thought the equals sign could go anywhere. Seven raised their hands.
Jimmy said, "I have an objection. What if the problem was just = 141?"
Somebody said that wouldn’t work because that isn’t a whole problem, part of it is missing. Then someone else said, "But what about Nate’s problem, does Tom agree with that?"
Nate’s problem was 141 = 141.
Tom said he agreed with Nate’s problem because 141 is the same thing as 141. I noted that Nate’s equal sign was in the middle and asked, "How is Tony’s problem different from Nate’s problem?" Tony pointed out that he had two equals signs.
I erased the = 141 at the end and asked, "Now how is it different?" Sandy said it meant the same thing as Nate’s, just longer. Several kids agreed with her.
"Tom, my first part of the problem is 141 and my second part is 141. It is like Nate’s," said Tony.
Tom said, "Now I am not sure."
We left the discussion at this point because Tom didn’t have any more questions and seemed to want to rethink things.
Equality has become a favorite subject of daily discussion in our math class. A student will say a problem and then ask if we can discuss it. Here is what happened this week.
On Monday, the problem was:
147 + 1 – 1 = 147 + 2 – 2 = 147 + 1 + 4 – 1 – 4 = 147
Jimmy said he wasn’t sure if he agreed. Anna said, "I don’t know if I agree because of the three equal signs. I don’t know if there can be three equal signs."
Tom said, "I agree now with these kinds of problems. The = can go anywhere, but all the parts have to be the same. In that problem, all the parts are the same."
Becky said it didn’t seem quite right to her. "It is like it is two problems, not just one." She went on to try to explain, but we didn’t quite get it. Then our college student went up to the problem and asked Becky where the first problem ended. She told him it ended after the first = 147.
He asked her where the second problem began. She said, "Where it says + 2 after the first = 147."
"So," he said, "the second problem is + 2 – 2 = 147?"
She said, "Wait a minute, that can’t be right. I see — the 147 goes with the + 2 – 2. Okay, I agree now."
On Tuesday, two students who have never given problems with more than one = gave these:
148 = 148 – 1 + 1 = 148
148 = 148 + 1 – 1 + 4 – 4 = 148
Everyone readily agreed and had no problem with either one. I was thrilled that they came to a consensus. I was also glad that our college student saw how much more powerful the children’s discovery was when they made it on their own. I pointed out that even when they didn’t all agree, none of them asked me what the correct answer was. What I wonder now is where to go from here. Any suggestions for a problem I could propose next?
Editor’s note: Gregg McMann has already e-mailed a suggestion to Holli, so stay tuned! Better yet, join the listserv! What’s above is representative of the pertinent issues discussed among participants.
To subscribe send an e-mail message with the words subscribe EXXONMOBILTNT in the body of the message (not on the subject line). The account from which you send your message will be subscribed to the listserv. Please address the e-mail to majordomo@math.byu.edu. After subscribing, you’ll receive more information.
Many thanks to Tevian Dray, Professor of Mathematics at Oregon State University, Corvallis, OR, for allowing publication here of his article. It originally appeared in "OCEPT Update," (Volume 5, Number 2), the Spring 2001 newsletter of the Oregon Collaborative for Excellence in Teacher Preparation. Subsequently it appeared in the newsletter of the New Mexico CETP. Ed.
In mid-March, I attended the Fourth Annual Mathematics Education Institute at New Mexico State University (NMSU) in Las Cruces, NM. It was hosted by Pat Baggett and sponsored by the New Mexico Collaborative for Excellence in Teacher Preparation (NMCETP), an NSF-funded project. The institute consisted of two days of workshops, followed by classroom visits to teacher preparation courses and to local schools. These courses are attended jointly by practicing teachers (who receive graduate math credit) and undergraduate preservice teachers. The activities are tried out with children in mentoring teachers’ classrooms.
The workshops presented materials used in the teacher preparation courses, aimed primarily at the K-8 level. Almost all of the workshop activities were hands-on — participants were active learners throughout. Several activities involved non-standard uses of simple calculators — including a method (based on continued fractions) for finding the rational number with a given decimal expansion. One of the more memorable activities involved using a series of mirrors to see an image in a given location, discovering that constructing perpendicular bisectors was a much more successful way to locate the mirrors than trial and error. All of these activities spoke to the theme of the institute, which was "Breaking Away from the Math Book", based on materials of the same name that have been developed by Pat Baggett and Andrzej Ehrenfeucht. Copies were distributed to participants. (See the New Mexico State University web site. Ed.)
In addition to these activities, there was a half-day mini-conference during which participants had the opportunity to discuss innovations occurring at their home institutions. I spoke about my vector calculus project at Oregon State University, emphasizing its possible impact on the teaching of related material at an earlier level, such as the basic properties of vectors. (Visit the Oregon State University web site. Ed.) Others participants included university faculty from several states spanning the country as well as a middle-school teacher from Manitoba, Canada.
The conference was exceptionally well organized. To say that participants were made to feel at home would be a considerable understatement. Discussions continued over dinner, and there was even time for some early morning hiking and, later in the day, some spectacular sunsets. But even more than the joy of simply talking with other like-minded educators, the dominant impression was of the success of the partnership between NMSU and the Las Cruces, NM public schools. It is worth pointing out in this regard that New Mexico has the highest percentages of both Hispanic students and Native American students of any state in the country. All in all, it was an enjoyable and energizing conference.
A fifth Institute is planned for next spring. Please look for information in Intersection next fall. Ed.
The Thirteenth Annual Meeting of the ExxonMobil Foundation/NCTM K-5 Mathematics Specialist Program will convene September 20-23 at the corporation’s headquarters in Irving, Texas. Regrettably, due to space constraints, attendance must be by invitation only. Invitations will be sent out in May to those who will be representing the various projects.
Helping to plan the meeting this year are Bill Fisher, Chico, CA; Vandi Hodges, Ashland, VA; Donna Little-Kaumo, Albuquerque, NM; Charlotte Stadler, New Rochelle, NY; and Bonnie Tank, San Francisco, CA.
Would you like to put your hands on some fine summer reading — for free? Just volunteer to review one of the two titles below for this newsletter, and the one you choose will be mailed to you soon — gratis. Interested? Please contact me. Ed.
Recently updated to support teachers as they implement NCTM’s new PSSM, the book Thinking Like Mathematicians: Putting the NCTM Standards into Practice is available for review. Written by Thomas Rowan and Barbara Bourne, and published by Heinemann, the 160-page paperback includes discussion about the new standards, how to put them into place, modes of assessment, and a series of questions and answers to help explain recommended classroom practices. See Heinemann's web site. Ed.
There are also two copies available of a new title by Grayson Wheatley and Anne Reynolds. It is called Coming to Know Number: A Mathematics Activity Resource for Elementary School Teachers. Published by Mathematics Learning, this book is a fresh and effective approach to number development that encourages students to construct meaningful methods of computation as well as enhance their mathematical reasoning. Please see Linda Coutt’s "Reflection." See the Mathematics Learning web site to learn more. Ed.
Mathematics in the City has announced the 2001 Summer Institute. Sessions will be held August 13 - 24, 9 AM - 4 PM, at the City College of New York.
The institute is a two-week inquiry workshop on teaching and learning mathematics. Teachers are asked to be learners in a mathematics environment where math is seen as the posing and solving of problems the searching for patterns, and the construction of formulae, models and algorithms.
The instructional staff is comprised of faculty from the City College of New York, the Freudenthal Institute in the Netherlands, and a team of Mathematics in the City teachers.
The cost is $800 per participant. Teachers can earn three graduate credits. Applications must be received by June 1.
To learn more, please call Sherrin Hersch at (212) 650-5091 or Herbert Seignoret at (212) 6550-6346.
I’m sorry that I inadvertently left out of last month’s issue the names of several newsletter recipients who were presenters at NCTM’s Annual Meeting. What a modest bunch you are, too! Not one of you called my mistake to my attention.
Nonetheless, I hope the presenters listed here will accept my apology — and I hope that there’s no one else I’ve left off that I still don’t know about! I’m sure that each of you presented a wonderful session: Eliza Berry, Bill Fisher, Pamela Geasey, Chris Horne, Jennifer Lara, Miriam Leiva, and Grayson Wheatley.
"Filling the Education Pipeline," an opinion editorial that originally appeared in The New York Times and The Washington Post on April 12, 2001, and in The National Journal on April 14, 2001, describes two of the Foundation's initiatives in mathematics and science: the K-5 Mathematics Specialist Program and the New Experiences in Teaching (NExT) program. You may read the oped on ExxonMobil's web site.
That you’ll send me somethin’. What’s going on with you and your project? Readers want to know!
Please send your contributions by Monday, May 21 to Jean Ehnebuske, 105 Hideaway Cove, Georgetown, TX 78628; phone, (512) 869-1580; fax, (512) 869-8477; e-mail, jean@intersectionlive.org. Many thanks to all who made this issue so meaningful. A special thanks to my husband, David, for all the photos that appear in this issue.
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