UCDS Spark Magazine #4, Spring 2008

spring 2008

Focus on Math

Partnerships: National Science Foundation Grant

Math Vitamin

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Parallel Lines: A Conversation with UW’s Selim Tuncel Give Me Some Space

In this Issue Sticky Curriculum 2 Spark is published by University Child Development School.

Math Vitamin®

People Who Inspire Us

Paula Smith Head of School

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Melissa Chittenden Assistant Head of School Teacher Education Center Director

What Works 12

Editor Jack Forman

Parallel Lines: A Conversation with Selim Tuncel

Math Partnership

Creative Fusion

Spark Logo & Publication Design Kelsey Foster Communications & Public Relations Director

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Contributing Staff Julie Kalmus, Ginger Goble, Brooke Leinberger, Diane Chickadel, Leanne Bunas, Kai Toh, Charles Kapner, Cory Goldhaber, David Garrick, Susan Foley, Natasha Rodgers, Angie Manning Goodwill

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Give Me Some Space

In Each Issue Greetings from Paula Spark Plugs UCDS Mission Statement

Photography UCDS Faculty and Staff For submission information, please contact Brooke Leinberger at brookel@ ucds.org. The editor reserves the right to edit and select all materials.

© 2008 University Child Development School.

All rights reserved.

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p 12

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One size fits all? It’s not likely. Whether examining a medication’s effectiveness, a child’s physical development, or our performance on virtually any task, we expect to find significant differences between each person we study. Increasingly, research data tracking these differences is becoming critical to decision-making in nearly every field used to both predict and to map our individual responses to the world. Most of us would not try a diet or treatment plan unless it was based on the latest research. Even then we would want to be carefully monitored to ensure that we were responding as expected. We take for granted that even the best “new thing” to hit the market will be right for only some of us.

It is surprising then to enter a typical K–12 school and find it organized as if all children were the same. Rather than the nuanced data-driven decision making that we take for granted in every other field, a student’s progress in most schools across our country is measured by a single high stakes test. We also find that teachers at every level are typically asked to address the wide range of differences in student ability, learning style, physical development, culture, language, and experience with little, if any, targeted data or professional support to do so. The test scores and school rankings that overshadow our national conversation about education keep us focused on outcomes but have not helped us to understand why one child may struggle while another does not.

The “Math Wars” that now dominate the K–12 educational airwaves mirror this problem. On the one side are “traditionalists” who see low scores in math as a result of unfocused “fuzzy” reform math and on the other side are the “reform math” advocates that see “rote drill and practice” instruction as the root of poor math performance. Both sides of the conflict point to the fact that students at every grade level are scoring poorly on state, national and international math assessments.

When looking at test scores in math it is easy to understand the alarm. In 2006–2007 for example, only 58% of Washington State’s fourth graders could meet the state math standards and in tenth grade only 51%.

Even our high scoring students are falling behind when compared to students in other countries. “Students in 11 out of 15 countries in the developed world scored higher than U.S. students in advanced mathematics.

No country scored significantly lower.”1 In a recent interview, State Superintendent of Public Instruction Terry Bergeson noted, “The whole country has been in denial about mathematics and now we’re sort of at a second Sputnik moment.”2 Perhaps, we need a “Sputnik moment” to push us forward. In order to ensure that we are actually moving in that direction, we would be well served to move the debate beyond the data that is currently fueling the Math Wars. The expansive developments in the sciences and in business are based on exhaustive research to discover not just the whats of our world but the whys. In 2006 a little more than 1% of the $107 billion federal budget for research was directed towards education, and this includes research focusing on higher education and adult education programs. We should demand that a conversation about math education be driven by the same level of research as is commonplace in every other field and that this research give rise to more targeted assessment tools, teaching strategies, curriculum support, and professional development for teachers.

It may be that our national effort to provide equity through the No Child Left Behind legislation is in reality keeping us from moving ahead by promoting a “one size fits all solution” that is crafted to address the lowest performing students. Equal access to math education will require us to structure schools and classrooms differently…as if children at every ability level could benefit from a stretch and could learn to love math. It is our experience at UCDS that this possible.

Sticky Curriculum®

by Rick Kirst and Gretchen Morse, UCDS Faculty

Math Vitamin

®

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ow do students at UCDS strengthen their analytic thinking and problem-solving muscles? With Math Vitamins, of course. These complex, open-ended story problems are a staple of the UCDS mathematics diet. Math Vitamins are treasured for the opportunities they provide for the children to apply their understanding about math concepts and to practice math skills in interesting and challenging ways. The Math Vitamin storylines pique the children’s curiosity by linking mathematics to themes and topics taken from throughout their school day. Whether Math Vitamins arise out of science, music or PE, they draw in even a reluctant student, connecting the math activity to the child through either an academic strength or an area of high interest.

There is a distinctive buzz in the room at the beginning of a new Math Vitamin. The children gather at the white board or large chart paper where the Vitamin is written and illustrated, to read closely and critically. While Math Vitamins exist at all grade levels, there are differences in the routines to accommodate differences in age, ability and experience. Children who are not yet independent readers, typically read the Math Vitamin with support from their parents, teachers and peers. Developing readers are supplied with their own printed copies of the Math Vitamin and are encouraged to ask questions of their classmates and teachers for help. Independent readers practice rephrasing the text in their own words while others record their summarization of the salient points in their Math Journals.

The details of Math Vitamin look different for different age groups, but the basic program is the same across our many levels. Let’s look in on a second and third grade classroom to get a feel for it: This morning’s Vitamin comes from the classroom read aloud book, The Invention of Hugo Cabret by Brian Selznick and references the wind-up toys sold at the toy booth:

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Hugo has been sneaking parts away from the toyshop to make his own inventions. Now it’s your turn to invent and build a mechanical windup toy – the type of small toy that might have been in the train station toyshop. Build a simple toy that uses at least one of each pattern block – yellow hexagon, red trapezoid, blue parallelogram, and green triangle.

What is the total value of your toy if the cost of a green triangle is 5 cents? Be careful as you decide on your final design, as you have no more than $3.20 to spend. Record your toy design and an equation for the cost.

As they eagerly read, the children identify the question(s), distinguish pertinent data, paraphrase summaries of the information, and explore approaches to a solution. After their first reading of the Vitamin, they begin to sort through the steps of their plans. They confer with each other and offer advice: “If the green pattern block is worth 5 cents, what are the other blocks worth?” “I don’t think you’ll need that many pattern blocks. Our wind-up toys have to fit on a regular sheet of paper.” “The red block is a trapezoid…what’s a trapezoid?”

Then, in a flurry of words, manipulatives and paper, kids get to work!

Working arrangements vary. At times, the children start Math Vitamins independently. After deciding on an approach to a solution and getting some initial results, they informally share and compare their progress with classmates. At other times, the children start work with a partner or in a small group. Teachers actively encourage these conversations. This variety of collaborative experiences supports the children’s becoming familiar with many different partners and groups and with different kinds of thinkers.

Some children return to the tables and with pattern blocks in hand, begin to build. The blocks are one of many manipulatives (see inset) used at UCDS across all grade levels and strands of math. An integral part of Math Vitamins, manipulatives allow the children to form concrete representations of the mental models they envision. In effect, manipulatives make thinking visible for all to see. In the same manner, the pattern blocks help a child think through a problem and literally see whether his or her solution works.

These same pattern blocks might also be in use in the Half-Day Classes to build shape and color patterns while in the 4–5’s, they are used to examine the fractional relationships between the blocks.

Continued >

As the children get farther into the Math Vitamin, the plans for solutions are as varied as the individuals who engendered them. Throughout the process, you can see the extent to which a Math Vitamin encourages children to develop systems and routines to support their thinking through a complex process. Some children start out with the pattern blocks, building their windup toy with green triangles. While a couple of them build, the others work on trading out triangles for pattern blocks of equal value. Another group begins with a budget in mind and calculates the value of each block, estimating how many of each kind they might be able to use and still stay within their budgets. Across the room, yet another pair is busy constructing a chart showing the value of multiple sets of each shape of block, anticipating that this will speed their computations later in the process.

At the end of the Math Vitamin session, the students join together to process their experiences. During this Math Circle time, they give each other feedback, ask questions, and share the methods and insights they gleaned as they solved the Vitamin. Most have an answer(s) that satisfies them and now have the leisure to reconsider and rethink their solution. Often during this reflective period the children see aspects of the Math Vitamin unnoticed the first time through or else see new possibilities in light of another’s work and build off the thoughts of their collaborators: “I never thought of it that way. I just started building. But I like how you figured out the multiples of 5¢ and then picked exactly the blocks that would equal $3.20 and built only with those.”

Others begin to notice methods that are more efficient and less laborintensive than their own and compare strategies: “Chunking numbers is way easier than how I did it. Thanks for showing me!!”

Still others are refining geometric observations: “I learned that you can make a hexagon with three blue parallelograms!”

Some Math Vitamins are written to fill only one stretch of work time. Other Vitamins are more complex and meant to last for days. One way to extend a Math Vitamin is to include a “Brainstretcher” or challenge problem(s) that are related to, but go beyond, the main Vitamin. There were several such extensions to the Hugo Cabret Math Vitamin mentioned earlier: One was to calculate, given a new set of values based on a length of time for each type of block, how long a wind-up toy would run on one winding. Another extension was to figure out how far the toy could travel in the time it ran on one winding.

There’s no end to where a Math Vitamin can take you

As students progress through the Math Vitamin they delight in testing out their ideas with each other, collaborating with one or more of their peers.

This as-needed, child-directed interchange provides opportunities for the children to hear and see how others envision the challenge. With fresh ideas in place, children tackle unexplored avenues of thought, drawing on their newly expanded repertoire of problem-solving strategies.

The children routinely turn to a variety of manipulatives for support as they puzzle through the Math Vitamin, building concrete representations of their plan and demonstrating their understanding. Manipulatives are routinely used to construct patterns, calculate with larger numbers or employ operations they are not yet feeling confident about such as the subtraction problems in Hugo Cabret that involve trading.

As the children work, teachers move from child to child to assess understanding of concepts and skills, from math facts to pattern recognition to geometric translations. Through these one-on-one interactions, as well as group sharing-sessions and reading through the math journals, teachers gain insights into the development of each child as well as the class as a whole.

These on-going assessments of the Math Vitamin guide teachers in the selection of other math activities and skill practices to be used in other parts of the school day or as materials for homework.

Student performance also determines the pacing and duration of the activities; adjustments to a Math Vitamin schedule and content are easily made to accommodate unplanned interests or discoveries—a logic problem can segue into an exploration of combinations. The students themselves often propose further related mathematical explorations and studies (“If there are twelve pentaminoes, how many hexaminoes would there be?”).

The Math Vitamin is a living thing that can change and grow throughout the time allotted to it. Whether a student or a teacher initiates the change, chameleon-like Math Vitamins can transform to meet the child at his level of understanding and to stretch and challenge the most highly developed mathematicians.

There is no trick to the creation of a Math Vitamin

It just takes time, teamwork and a commitment to the process that UCDS teachers have followed for years. The first step occurs during the grade-levelteam planning session where teachers set the details for the following week’s instruction. Using the information compiled from the informal observations and assessment of the year’s math work, teachers target which math strand, number operation(s), and manipulative(s) will be incorporated into the Math Vitamin. Teachers use the UCDS Math continuum as a checklist to ensure that Vitamins cover a variety of skills and strands over the course of a year.

Once the mathematical objectives have been identified, the real fun begins. No part of the curriculum is exempt from becoming part of the storyline. Over the years, Math Vitamins have asked students to design courtyards with the Greeks, expand recipes to feed several classrooms, plan and budget for cross-country trips with characters from read aloud or literature book characters, explore the solar system, triangulate with Pascal and dance with Fibonacci. Developing the story line is cooperative creativity at its best. As the ideas fly, they are woven together into stimulating and challenging problems that intrigue and engage students.

Teachers at UCDS learn how to write Math Vitamins on the job by being part of a team that creates and manages them on a daily basis. However, all teachers at UCDS have gotten their start by participating in the weeklong math workshop offered each summer and attended by educators from independent and public schools. Teachers new to UCDS are required to attend for their first two years, and many of them return to participate in the course as instructors.

Last summer the teaching team created Math Vitamins to begin each day of the workshop. Because the final installation of the Harry Potter series had just been released, it was decided that the theme for the week would be centered around various aspects of the wizarding world. To begin each morning the participants delved into wand making with unit fractions, exchanging dollars at Gringots and creating geometric banners for their Quiddich teams. Dressed as wizards in capes and costumes, teachers experienced first hand how math activities linked to a popular theme can turn math into a joyful experience.

As teachers gain experience, they bring new ideas and a fresh perspective to their teaching teams at UCDS. Working with a new theme each year, these teams develop the story lines that are woven together with the math concepts to create Math Vitamins. From year to year the math concepts remain the same, but the context that they are written in is refreshed in a manner that is highly motivating for students and integrated into all areas of the curriculum.

Each year, when fifth grade students graduate from UCDS, teachers routinely ask them, among other things, which subject areas were their favorites and why. Math Vitamin consistently is at the top of that list. A former UCDS student with a knack for written expression sums up her experiences. “One reason why I liked the King’s Plaza Math Vitamin is it included a lot of designing and drawing and I enjoy both. In this Math Vitamin learning that one of my strengths in math is geometry completely changed my perspective.

Math is now one of my best friends.� s

Manipulatives come in a variety of shapes, sizes, colors and materials. These objects provide opportunities for children to construct mathematical knowledge.

The children maneuver these materials to build models of their thinking and to demonstrate for others their ideas and understanding of math concepts.

TYPES OF MANIPULATIVES: Base-ten blocks, Multi-base blocks, Attribute blocks, Multi-links, Color tiles, Cuisenaire rods, Geo-blocks, Geo-boards, Tangrams, Discover It boards, Alpha Shapes, Fraction Tiles, Polydrons

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A recent Math Vitamin studying fractions done by a student in a first–second grade class.

A Math Vitamin about budgeting for a feast done by a student in a fourth–fifth grade class.

People Who Inspire Us

Parallel Lines Finding Congruency Between Elementary and Collegiate Math Programs: A Conversation with Selim Tuncel

One of the benefits of serving a diverse group of families, is sometimes you find that the best resources are already a part of your parent body! As we prepared for this math-themed issue of Spark, we once again found this to be true.

We heard from Selim Tuncel, chair of the Math Department at the University of Washington (and the father of two boys enrolled here at UCDS) that some dramatic changes have recently occurred in the program he oversees. Their recent move toward smaller class sizes and an experiential learning model in entry-level Calculus courses has resulted in a two-fold increase in the number of declared math majors and increased student and faculty engagement in the math community.

As we talked about what has been an eight year effort to alter the math program at UW, we felt almost as if we were looking into a mirror: we found many parallels between our programs, most importantly the emphasis on teaching math in an engaging way. What follows are excerpts from our conversation.

SPARK: We’ve been talking a lot about the changes that have happened in the math program at the University of Washington. The story is so compelling! It seems like it started with some outof-the-box thinking in terms of how you taught your entry-level calculus courses.

SELIM TUNCEL: Before we started, the classes were quite large and traditional. We’ve moved from the idea of lecturing students who sit and absorb material to thinking about how can we involve them more?

How do we help a broader range of students to get engaged?

It took four years of planning to get it done. The first year was the last year of Doug Lind’s chairmanship, and he appointed a departmental committee to go talk to the faculty and students and client departments just to go see how we’re doing. And that kind of made it clear that there was a lot of room for improvement.

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And that was motivated because members of the faculty were feeling like the program wasn’t succeeding? Or was student feedback a part of this too?

I think, in general, both student and faculty feedback were part of the change. Actually, that year, we sent out course requests to the faculty, and only one or two people out of 50 or so professors actually requested to teach calculus!

That’s like enlisting!

Yes, I think that was a pretty good sign of where we were!

So the next year, one of my colleagues, Paul Smith, went around and talked to other large math departments, other places our size in state universities to see what they were doing. Some had smaller classes and others emphasized more experiential learning. There were all sorts of models.

students get stuck and T.A.s don’t give them the solution. They give hints.

They coach! It’s like what we do, every part of this.

It’s a Math Vitamin!

It’s very similar, actually. Students meet together and their small groups have a chunk of time in the middle where they work on these problems.

And at the end, there’s enough time to go over the problems where different groups present what they did. “Here’s how I solved it.” Or, “Here’s how far we got, but no further.” If there’s enough time to say, “Okay, maybe that’s not a dead end. Let’s see if we can go from there to where we want.”

Students work on it together, contributing as a team.

What do you mean by experiential learning?

Professors teach three days a week: Monday, Wednesday, Friday. And the class breaks up into smaller quiz sections on Tuesdays and Thursdays.

In the old model, someone stood in front of the board and it was almost recitation; in the extreme case, a professor or TA wrote out the solution and the students simply copied it. In the good cases, of course, there was interaction, but the model was not built to encourage interaction.

So part of the effort was that a number of my colleagues developed worksheets. To test them out, we extended one of the quiz sections from 50 minutes to 80 minutes. That gives enough time to teach a worksheet. Students gather in small groups of four to five, the T.A. goes in, and introduces the worksheet, which is related to something they’ve done. The worksheet is designed to have a couple of stumbling blocks, a couple of places where most people are supposed to get stuck and then discover something.

Every week, the T.A.s get together in groups to discuss the worksheet, and students’ response to the previous worksheet.

So all of the T.As understand that here are these huge two stumbling blocks that the students are going to come up against.

Exactly. And you don’t just give T.A.s the answer to the worksheet. It’s a discussion of how students got around the problem in the past, how they solved it and what to expect as a reaction this time. So the

And obviously, there will be different levels of understanding and processing speed in that group, and also probably completely different ways of picturing it.

Continued >

In memory of

John Neilson UCDS parent, John Neilson loved ideas; those he found in literature and those he gained through a deep appreciation of world culture, math, science, art, music, philosophy, and physical excellence.

In 1999, at the age of thirty-eight, John lost a hard fought battle against non-Hodgkin’s lymphoma. In honor of John’s life, The Neilson Endowment Fund was created. Through the Teacher Education Center at UCDS, we use this endowment to create and share programs that offer children access to big ideas.

John was an inspiration to us in life and we dedicate this, ‘People Who Inspire Us’ section to him.

There’s also the whole math anxiety thing that we all have to varying degrees. Working in groups, you break that barrier somehow. Math is somehow associated with being smart: “Am I going to look dumb if I say what I’m thinking?” People quickly learn that, no, there’s no magic to it. Just almost everyone adapts: it’s not a big deal.

Once you start enjoying it, the rest comes.

Another thing I haven’t mentioned is The Math Study Center. All the instructors, professors and T.A.s, hold their office hours in there. There’s a section of the Math Study Center that’s dedicated to each course, so any student who happens to be there with a question will find a T.A. or instructor that they can go to. It’s a way of making help available, not all the time, but close, whenever the place is open.

It seems as if suddenly the mathematical process itself would become more transparent as people work in groups. Do you think that’s been a change?

The help itself seems so important, but also just the statement that makes that you’re committed to the community.

Yes, that’s definitely been a change.

If you’re going to appeal to a broad group of students with very diverse and different backgrounds coming from all over the place, different high schools, different economic backgrounds, everything; you really have to give them a chance, basically. If you don’t, many of the people get intimidated.

Another part of our change relates to the kind of discussion that’s been raging elsewhere and hit our state last year about ‘Math Wars.’ It’s basically about whether you teach the techniques, or do you let them discover for themselves?

When I learned this stuff in the ‘70s, there were standard texts and instruction was mostly about learning technique. And then we went in the other direction to word problems, maybe even a little too far.

Most people now see that it needs to be a combination. That’s something else we sort of strived for and hit, that you know, you have to give students the technique and the methods and the formulas to a reasonable extent. But the methods and the formulas alone can be daunting, something that the students can’t connect to. You have to give them the experience of connecting and applying.

And we’re always wondering how it’s all working. What are instructors’ experiences? Where are their students getting stuck? Where are they having difficulty?

That you’re available, exactly.

And also it may allow you to see a different viewpoint and different style of instruction than that of your instructor or T.A. That slightly different approach may be just what clicks for you.

Large classes don’t create the right atmosphere for this kind of interaction and creativity to take place. Before we started, the class size was probably 160 students, so we halved that to 80. Actually, 81 because 81 is divisible by three.

This is exactly what we do! You’re looking at the effectiveness of what you’re doing and continuing to make tweaks because the students obviously change, and the teachers are changing as well.

The tweaks really keep you going. But I think after a few years, you need to do a more major revision because tweaks can take you only so far. My colleagues tell me we’re going to hit that point in a couple of years: they’re already seeing signs of things getting a little stale.

What other kinds of changes have you made?

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It is the math department, after all!

Hah! Now, 81 breaks into quiz sections of 27, which gives you a reasonable size. Actually when we did it, the ideal size was 24 because it was six groups of four. So we didn’t quite hit the ideal, but it’s close enough.

The lecture’s different as well. I taught one of the first pilot sections of 80 students the very first year we tried it. Forty would be better, but 80 is doable; it’s about the upper limit. You can stand in front of a class of 80, and have an idea of what’s going on and make eye contact with most of the students.

So it’s more of a two-way relationship.

Yes, the lectures are much more interactive.

In a two-year period, soon after we implemented this, our math major enrollments increased dramatically.

We went from about 150 math majors to over 300!

One year, it was 150, the next year it was 250, then it was over 300. And we’re sort of stable now, about 320.

What do you chalk it up to? What is the biggest difference?

It’s that students now actually enjoy it.

The professors enjoy it because, as you know, communication is a more of a two-way street.

Everyone has a good time. And we’re teaching 16,000

University of Washington Math students are encouraged to connect with teachers, work collaboratively, and build community by visiting their math class’ area at the Math Study Center.

enrollments a year. In the fall, we were teaching 4000 students in calculus and pre-calculus, so it only takes a fraction of the 4000 to say, “Oh this math wasn’t that bad, I can now take the next one.”

And students see other students doing math and succeeding and appreciate this. Math majors have been winning awards, such as Rhodes and Marshall scholarships and six Dean’s Medals in the past eight years. It sets an example. It gives entry-level students something to aspire to.

So really, it’s a community. It’s no longer autonomous people working on math.

Yes, it is a community, much more so.

Can we talk about the change process itself a little bit? How were you able to implement all of this?

The critical thing to know is that it was and is a department effort; I’m just one out of 50 or 60 people.

We all worked on it as a department. It never would have been possible without the great majority being on board.

All good projects are like that!

Eight years, it took to do this. As I said, the first year was consensus-building. The second year, one of our colleagues went out to universities and actually talked to people, found out what they did, came back

and reported to us. Our original idea was that at the end of that second year, we’d decide what to do based on this information.

How and what to change.

Yes. The third year, we formed another committee which was probably the most intense. Lots of department meetings, people discussing, heated and creative discussions, helpful discussions. Our Chair at the time was Don Marshall. He was constantly communicating with the Dean to see if all of this was feasible.

The next year was discussions with the Dean and the University to get funding. So that was the start of a four-year project, which was supported by a fund called Tools for Transformation, which was centrally funded.

The first year, there were some glitches, and we’d talk about them and actually some of them had ridiculously easy solutions. It would look like a problem and we’d try something, and hey: the problem’s gone!

Before all of this, we used to think, “We teach well.

We teach the content. We’re doing our job. The rest is sort of presentation, sales, et cetera.” But we learned that how you present the material really is important.

We learned that the payback from making sure the student satisfaction is there is huge. It just creates an atmosphere that’s positive and better for everyone. It enhances learning. And again, you of course know this very well in this school. s

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What Works

by Julie Kalmus

WORKING WITH A PARTNER BUILDING A UNIQUE THREE-SCHOOL MATHEMATICAL PARTNERSHIP

All around the school, there’s an energy.

In one room, a group of half day preschool students are sitting at tables excitedly discussing pizza recipes for rocks and cutting circular pizzas into halves, thirds and quarters. A three-foot tall, three-year old jumps on her chair and proclaims, “Chair announcement! I think I might have just done a little bit of fractions and it’s all about parts and wholes!”

Across the campus, in a first-second grade room, students are offering ideas for things they can teach for Kids Teach Teachers Day. A girl raises her hand to announce, “I kinda love math. Could I teach that?”

The circle nods in agreement and proceeds to argue over who will get to write the day’s math vitamin.

Meanwhile, in a fourth-fifth grade class upstairs, students clamor over each other to explain their understanding of Pascal’s triangle to visiting guests.

“Pick me!” a smiling girl pleads, pencil and paper in hand.

This isn’t a special day: you can hear this kind of enthusiasm each day in the halls of the school.

Our students’ perpetual mathematical joy reflects a journey to build a math program that is as deep as it is inspirational! It’s a journey that began almost twenty years when the school was still growing.

True to its roots as a lab school, UCDS has always been interested in teaching math well. Even twenty years ago, students were graduating with high levels of competency, having worked throughout their school careers with creative teachers. Manipulatives and in-depth problem solving were already common.

But the kid-driven excitement for math wasn’t there.

Our faculty were inspired by the potential they saw in building a hands-on, inquiry based program, a community of math exploration where our students’ excited questions led the charge. We saw the promise of joyful learning and students empowered by authentic challenges to their creative intellect.

But how do you do this? Our desire for something more became a catalyst for innovation.

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First, we read.

Howard Gardiner had just unleashed his work on multiple intelligences and we were propelled to look at teaching math to different styles of learners.

Heated conversations about how to honor the artistic and global thinkers in our school ensued.

Grade level teaching teams sprawled out on the floors of their classrooms and spent hours working to write our first Math Vitamins. We grappled (and still sometimes do) about how to make questions open-ended and about how to meet learners at so many different levels at once.

Other math experts found their way to our bookshelves and inspired more work. We revisited Marilyn Burns’ work and incorporated math games into our curricula with greater depth and context.

Constance Camii and a host of other theorists drew our attention. Our math community was already changing. And it was exciting!

Next, we invited math experts into our community to help us create a math program that still seemed largely theoretical. We spent valuable in-service days exposing us to other methods of math instruction and challenging us to develop a more hands-on and inquiry based approach. During week-long workshops, we played with math manipulatives only to discover for ourselves the reasons behind the algorithms we had been teaching! Seeing, building and drawing the partial products in multiplication arrays or the visual proof for flipping and multiplying in the division of fractions blew our minds!

Then, like any good faculty must do, we played.

We created trains and animals out of Cuisenaire rods. By making our own spaceships out of Base Ten blocks, we came to understand why our students always did it. Just like our students, we held different colored transparent fraction tiles over our eyes to make the classroom look like the ocean or hot lava before we used them to explain common denominators. We built Unifix cube castles, multibase museums, and Geo-block towns.

We’d like to recognize and thank our partners in this project: The National Science Foundation Seattle Public Schools Loyce Adams, Kim Gunnerson, and the Applied Math Department at the Univeristy of Washington

And while we played and explored, we talked.

A lot. After a year or so of exploring together, it became easy to look at colleague and say, “Prove it!” when they announced an answer. “Show it,” became another common phrase as we used each other to deepen our own conceptual understanding of math. At the end of each session, we could not wait to work with our students. Almost immediately, we started teaching differently. Math Vitamins and manipulatives became a mainstay of our curriculum.

Meanwhile the faculty created a Math Continuum, a developmental tool to assess the diverse skills that our students were attaining in this new math model. Creating the Math Continuum forced us to confront assessment, benchmarks and beliefs about student learning; and by 1998, it had become a usable assessment method for every level in the school. Everyday, teachers talked about the lessons they had taught and brainstormed together how to deepen, extend and improve student learning.

Things were changing before our eyes! The excitement around an “a-ha” was palpable and everyone in the school felt it. Students started jumping on chairs to announce their math discoveries and ran into the building to read the morning vitamins with their families. And once kids began to catch the new math wave, we invited parents along for the ride, hosting our first Family

Math Night. We had created a shared ownership and understanding of our math program. Kids, teachers and parents had a passion for math. Done, right?

We actually thought we could go further.

When you are feeling passionate about something, you want to share it with others: We were no exception. We took our story on the road, presenting this journey at other schools. And our passion and program were getting noticed!

Educators from other schools started asking to come to us to see math in action. We were invited to present at the Pacific Northwest Association of Independent Schools, the National Association of Independent Schools and even the National Council of Teachers of Mathematics.

At the same time, Loyce Adams (Associate Professor at University of Washington’s Applied Math department and UCDS parent at the time) took Continued >

notice too. In her role at UW, she was consulting with elementary school math programs. In 1999, she initiated a dialogue with UCDS about the characteristics of strong math programs and, of more interest to her, strong math teachers.

It was a conversation that led to an innovative mathematical partnership.

Loyce had watched closely as UCDS transformed its math program. In her work at UW, she was interested in supporting public schools to build stronger math teaching. Loyce realized she had a unique asset at her disposal—graduate students with a passion for math and a deep understanding of its concepts. This brought up a core question: If graduate students taught math beside classroom teachers, would math instruction and student achievement improve? Her passion for supporting education led her to write a proposal to the National Science Foundation to launch what has become the GK–12 partnership.

Through the GK–12 partnership, graduate fellows are paid a stipend to work 10 hours a week in public school classrooms in Seattle. The graduate fellows help implement math curriculum but, more importantly, share their deep mathematical understanding with teachers. The GK–12 partnership was the first of its kind in the United States.

Initially, it proved challenging for graduate math fellows to work directly with elementary school teachers and their students. Though the graduate fellows had exceptional mathematical backgrounds, they did not have formal educational training. Often the reality of understanding classroom cultures and managing student behavior superseded the implementation of new math techniques.

The program really needed a third arm. Loyce looked to a new resource—UCDS teachers, who were passionate about teaching elementary math classes and comfortable individualizing instruction to meet the needs of a variety of learners. New questions emerged: If UCDS teachers partnered with the graduate fellows and the public school teachers, would there be greater chance of the team’s success? Could UCDS teachers not only add value to the project but also benefit professionally from the experience? The proposal of this partnership was intriguing. When the conversation reached the UCDS faculty, enthusiasm was high: Teachers were excited to experience a diversity of classroom cultures as well as new curricula. They embraced the opportunity to work with different students and

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teachers and the opportunity to benefit from the fellow’s mathematical acumen was also compelling.

With the support of the administration, six faculty members volunteered to participate in the project’s partnership with Thurgood Marshall Elementary.

Once a week, they would leave their UCDS classroom to co-teach a hands-on math lesson with one of the Thurgood teachers. Over the next seven years, we would also build relationships with Leschi and Emerson Elementary Schools in Seattle. In these schools, the program placed twelve members of the UCDS faculty to partner with classroom teachers each year. With UCDS teachers and UW fellows by their side, participating faculty were learning to better individualize instruction and give greater conceptual depth to their math students.

The opportunity to work in such diverse settings inspired in-depth conversations back at UCDS. The complexity of the partnerships demanded real discussions about ways to better support teachers and students. Similar themes from the evolution of our own math culture emerged: There was a need to create math communities and continuity throughout grades levels. Schools identified promoting a love of learning math, setting and meeting high expectations, increasing collaboration, authentically assessing student work and learning new methods of coaching as crucial to creating these communities.

Having identified these themes, UCDS worked to create spaces to promote open conversations among GK–12 partnering faculties. We began close to home by opening our week-long, Summer Intensive Math Workshop to all GK–12 participants. Objectives of the workshop included deepening conceptual understanding, exploring manipulative use, creating strategies for differentiated instruction, and giving faculties an opportunity to start the process of building math community amongst themselves.

Throughout the week, faculties got to test their beliefs about math instruction and explore new methods of instruction.

However, fun and passion were also a major focus, perhaps the most so. Facilitators infused the workshops with playful themes. Participants used the then-new Austin Powers films to explore fractions by creating life-sized costumes for Mini-Me (we called it The Spy Who Sewed Me!). To investigate geometry, they used three-foot long newspaper roles named “Loompa Logs” to create models for new candy factory wings during a Charlie and the

(top) Teachers visit UCDS each summer to take part in the Summer Intensive Math Workshop. (bottom) An elementary student from Seattle’s Thurgood Marshall Elementary School explores place value math with manipulatives.

Chocolate Factory week. Inspired by Harry Potter, participants used multi-base blocks to create a full scale Wizard’s Market. As groups worked industriously towards common goals, they also laughed and celebrated. As facilitators, we marveled at the familiarity of the conversations that faculties from partnering schools were having. The laughter and enthusiasm that pervaded the weeks at Math Workshop began to cross into the halls of their own schools.

We also created a format to add discussion and meeting time with the faculty and administration at each school. During the school year, we met regularly. Hands-on practicum, teaching partnerships and collaborative meetings provided opportunities for regular reflection on the actual work being done in the field. Here, questions became more practical, as we focused on implementation across a broad spectrum of classroom environments. Other meetings were dedicated for faculties to develop a shared vision for their school’s math program. Schools have used these sessions to set goals for building a positive

culture around math learning, to plan family math evenings, to create school wide math activities and themes and to share and assess student work across grades.

The ever-changing landscape of schools requires that with each year of this partnership we adapt, modify and refine. Meanwhile, we find that the gifts of the partnership also evolve. As we reflect on our own learning, we are grateful for the opportunity to share our knowledge and experiences with a broader community, to stay connected with our sister schools, and to bring back lessons from our partnerships to UCDS. This partnership is not a lecture or a book. It is a hands-on, responsive professional development experience for all involved. The journey we started twelve years ago continues as we further embrace the aspiration to innovate and learn beyond the scope of our own walls.

All the while, reminding us that innovation is something to encourage and celebrate! s

Creative Fusion Ellen Winner, continued

by Ben Chickadel

GIVE ME SOME SPACE A Blueprint for Connecting Math and Technology

W

e are three-dimensional beings. We interact with each other and our threedimensional world constantly. We elegantly and intuitively move through the X, Y and Z axes from an early age and are constantly affected by our positions in them. Simply put, we are hard wired for space.

But not until very recently has technology been able to truly assist and describe this experience.

Only within the last few decades have the designs of three-dimensional objects been aided by the

neighborhood. They measured angles in interior and exterior space with compasses, and calculated the area of the spaces using their standard-unit measurements of walls and floors. They studied architectural design and building an archive of spatial tools.

After creating blueprints and elevation drawings for their small structures, the third and fourth graders led mock architectural presentations with a panel of teacher clients to prove the feasibility of their designs. Once the plans were done, it was time to make simple paper prototypes, snipping, folding and taping cardstock to translate their 2-dimensional designs into 3-dimensional mockups. And, though excited to see their plans take form, kids wanted more. “Can we build one or two of these full size?” one student asked. “I wish we could see how these would look in real life.”

Enter Technology This Math Vitamin seemed like a natural fit for an exploration into Computer Aided Design (CAD). Kids had created two-dimensional depictions of their designs with the tried and true method of graph paper and pencil; now it was time to add some depth to their work.

But before they could really bring the computer to the designing table, they first needed to consider the nuances of space itself. In the technology classroom, we talked about how to form a space: How do you make it with your hands? Where’s the geometry in simple forms?

What changes, like folds, edges, angles, affect the overall space?

The exploration of space began by creating threedimensional form and space from two-dimensional materials like paper and cardboard. Using Microsoft WORD, each student created a simple pop-up template of their name, printed it out, cut and folded it specific ways to create a three-dimensional stand-up name.

When the card was unfolded, students witnessed the “action” of three dimensions. We also looked at various pop-up structures and paper sculptures.

Next, they were asked what does an unfolded cardboard box look like? And how could one be designed to fit a specific object or product? Using templates borrowed from packaging design, students folded and cut the patterns to create a three-dimensional form.

It was fun and challenging to guess what an unfolded two-dimensional layout would look like as a completed tree-dimensional box. As the students began this process, each time they folded and cut out a pattern

and came up with the three-dimensional form it was a surprise. They made lots of connections between this magical 2-D to 3-D process and their own 2-dimensional building designs. With some practice they were able to imagine the three-dimensional form just by studying the two-dimensional pattern!

We were creating a culture of spaces: how to create them, how to change them, how to talk about them. Now it was time to go digital!

The CAD Process Learning Computer Aided Design is kind of like re-learning how to use your hands. To embark on this odyssey, students were introduced to the CAD program Google SketchUp! SketchUp is an intuitive, easy to learn program, a great resource for schools and —BONUS— it’s free! Students can experiment with it to mock up threedimensional spaces, replacing the intuitive paper and scissor techniques with purely virtual manipulatives.

Kids learned to navigate SketchUp by learning commands and mouse techniques to create different 3D forms on the computer screen such as boxes with numerous facets, buildings with windows and indentations, and rooms with various details. They even learned how to place a virtual person into a space to better imagine the scale of what they created.

Once the students reached a high level of proficiency in SketchUp, they input the data gathered from their own pencil and paper blueprints. Once they entered these dimensions, kids could see and manipulate their basic house model and ask themselves how the space could be improved upon. Using the computer program as their design tool, they went on to create a deluxe model by adding on other rooms, furnishings, windows, storage, etc.

Taking their designs even further, they continued the process, adding on even more detail such as a custom deck or glass paned windows. The final stage of this project was for the students to imagine they had clients interested in purchasing their designs. They created presentations for these clients complete with animated walk-through tours and computer representations of children so the clients could see how a 4’ 2” person might move through the space. What would be at eyelevel? Would they have to crouch to get through the door? Could they stand comfortably and peer out the window? No detail was left out.

Continued >

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From the Drafting Table to the Computer Screen: Eleven Steps to Digital Design Over the course of several weeks, students found an exciting nexus between their math and technology studies and emerged on the other side with a new (and better) understanding of each subject area. Follow one student group’s geometrical journey from two to three dimensions!

Step 1: From The Ground Up First things first, the students started inputting the measurements from their math journals into SketchUp. And what better place to start but from the ground up? Kids entered only the length and width at this time.

Step 2: Going Up Next they pulled the base up to the full height of their tree house. This is that magic step—in one simple mouse drag and click, two dimensions suddenly became three before their eyes!

Steps 3,4 & 5: Raising the Roof Students had to first measure the line for the base of the roof (dotted line in fig. 3). This helps in the next step. It was a plus having all the measurements and diagrams documented for reference in their math journal: reinforcing the importance of carefully documenting their Math Vitamin work.

Students began to delegate jobs to one another as they progressed, “You navigate. I’ll read you the dimensions.

You type.” The Design Teams started to take shape.

They made a peak to the center of the cube to get an accurate height (fig 4). The final step in the roof was to carve away any unnecessary bits (fig. 5).

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Steps 6 & 7: Slight Adjustments At this point, the students had the typical house that everyone thinks of when someone says “house” (fig. 5) but this is a tree house with certain criteria and design considerations. Maybe the roof was a bit (or a lot) askew then they moved the peak (fig. 6).

Try that with pencil and paper! The same thing happened with the walls; This particular house had a slightly overhanging wall (fig. 7).

Drawing a line, they carved away the unwanted edge. Measuring, measuring, and measuring really helped during this stage of adjustments.

Step 8: Cloning The students initially started using SketchUp with the intention to virtually realize their original tree house. The technology enabled the students to extend the challenge and build on the Math Vitamin from their classroom.

In Technology class, they became designers for prospective clients. They all agreed that giving choices to their clients seemed like a good idea. They duplicated their base in order to have three working versions of their model: a base (already made), a deluxe version and, yes, even a super deluxe model.

Step 9: Deluxe SketchUp allows users to add textures, materials, colors and components. The students’ jobs were to find appropriate materials depending on what model they were working on. In the Deluxe model they added the necessities: entrances, windows, etc…

Step 10: Super Deluxe!

Most of the tools in SketchUp are things we have seen before, just super-sized. A pencil draws but it does a lot more, like sensing midpoints, endpoints and lengths. The class had to take a little time to (re)learn how to use some of the more advanced features. With a little experimenting and trial and error, kids were able to dream up some pretty amazing creations for their third and final variation!

Step 11: Presenting The last step was presenting the models to perspective clients.

Using the animation feature in SketchUp, students created fly-through animations to present to the class, the prospective clients. This was a great way to see each others’ work, discuss trouble-shooting techniques. Both the student architects and prospective student clients were simply awed by each others’ designs.

Meanwhile, teachers were amazed to see how one simple Math Vitamin could exist in so many domains and excite students on so many levels, all at once. It’s another reminder of the value of collaborating, the synergy that a curriculum can attain when we work to creatively fuse our ideas together.

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Knowing and Teaching Elementary Mathematics Teachers’ Understanding of Fundamental Mathematics in China and the United States

By Liping Ma Liping Ma, a Senior Scholar with The Carnegie Foundation for the Advancement of Teaching presents a cross-cultural examination of what’s working (and what isn’t) in elementary mathematics, particularly as she compares the competency of Chinese and American math students.

Ma reasons that the value of the number of years of schooling in mathematics pales in comparison to the importance of a deep understanding of the math processes themselves. Meanwhile, she presents an elegantly readable research-based account of where China and the United States stand mathematically speaking.

The National Math Panel Final Report President Bush created the National Mathematical Advisory Panel in 2006 to assess and assure future success in mathematics in the United States. Their findings, including recommendations for institutional practices, materials, professional development and assessment are available for viewing at: www. ed.gov/MathPanel

Weaving Your Way From Arithmetic to Mathematics with Manipulatives

By May Laycock and Peggy McLean Used by UCDS Faculty at all levels, this helpful assessment tool and curricular planning aide demystifies manipulative use and helps make dynamic curricula like Math Vitamin possible.

Laycock and McLean present an intuitive, concrete approach to math manipulatives appropriate for each mathematical strand, all the while encouraging students to ask not just the whats of particular math processes, but the whys.

Google Sketchup http://www. sketchup.com UCDS Technology Specialist Ben Chickadel wrote about this exciting tool and its concrete connections to Math Vitamins. Even our youngest students eagerly draw three-dimensional creations in the program, building technological facility as they excitedly create. It’s available for free download online.

Spark Plugs Mind, Brain and Education By imbes (international mind, brain and education society) Volume 1, Number 2; June, 2007 Florence Mihaela Singer contributes a remarkable article to this recently created journal, entitled Beyond Conceptual Change: Using Representations to Integrate Domain-Specific Structural Models in Learning Mathematics.

Her findings underscore the importance of elements we have found to be central to the successes in our dynamic, multi-domain Math Vitamin approach. Specifically, she finds that the more ownership that students have of an idea, and the more empowered they are to apply that idea in diverse settings, the greater their overall understanding and internal ability to create connections. We at UCDS were excited to see that our Math Vitamin approach was consistent with Singer’s findings!

Research That Matters: Does Modern Math Education Add Up?

By The University of Washington’s College of Education The University of Washington’s College of Education uses this, the fifth issue of their “Research That Matters” series to confront what has colloquially been known as the “Math Wars.” In our current “No Child Left Behind” era of hyper-focus on meeting standards and reaching the bar of such state assessments as the WASL, it’s difficult to know how exactly to help students learn mathematics and actually “get it.”

This revealing study of the current state of mathematics suggests that we take a new view of math, both by re-examining the traditional mathematical environment and asking ourselves, “what are we preparing students for?” Find out more at http:// education.washington.edu

National Council of Teachers of Mathematics http://www.nctm.org As faculty member Julie Kalmus writes in her article about our ongoing UCDS math partnerships, the NCTM serves math teachers, math educators, and administrators by providing math resources and professional development opportunities. Its web site offers resources for all age groups from elementary school through college and an ever-expanding community of teachers committed to teaching math successfully.

Paula references a statistic found in Research The Matters: Does Modern Math Education Add Up? by The University of Washington’s College of Education 1

The quote from Terry Bergeson is from Research The Matters: Does Modern Math Education Add Up? by The University of Washington’s College of Education 2

UCDS Board of Trustees Officers Eric Fahlman, Chair Bill Nicholson, Vice Chair Janet Donelson, Treasurer Nan Garrison, Secretary Members at Large David Bolin Elana Lim Eric Sanderson Greg Headrick Julie Petersen-Dunnington Julie West Prentice Kate Marks Kelly Webster Peggy Rinne Perry Atkins Roger Page Ex-Officio Members Mike Riley Joelle Harrison Paula Smith University Child Development School 5062 9th Ave NE Seattle, WA 98105 206-547-UCDS (8237) Fax 206-547-3615

The UCDS Mission University Child Development School is centered around the lives of children and is dedicated to the development of their intellect and character. We actively encourage, and the school everywhere reflects, the process of joyful discovery that is central to meaningful and responsible learning. Teaching is individualized and responsive to the talents of each student, and the curriculum is rigorous and integrates the concepts and skills embedded within the major disciplines. Our students are chosen for their promise of intellect and character and are selected from a crosssection of the community. Our faculty members are leaders in their fields, supported in advancing their studies and encouraged to share their knowledge widely.

In pursuit of these ideals, and in recognition of obligations beyond the school itself, we strive to be an innovative leader in education, serving as a model for others.

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