COMET • Vol. 3, No. 13 – 3 April 2002


(1) CAHSEE Mathematics Project Resource Guide Rollouts

Contacts:  Patricia Duckhorn –  (916) 228-2244; Victor Gee –  (510) 670-4538; Paul Pechin – – (916) 228-2631

Over the next two months, there will be eleven one-day regional rollouts of Standards-based resource materials focusing on the California High School Exit Examination (CAHSEE). The “CAHSEE Mathematics Project Resource Guide-Part 1” is the product of a statewide team effort headed by Sacramento and Alameda County Offices of Education.

The components of the Resource Guide include the following:

* Problem bank for all CAHSEE and related standards

* Adopted textbook citations for all identified CAHSEE and related standards

* Application lesson plan for each CAHSEE standard

* Map of related standards back to the fourth grade

* Staff development plan for district/site/department/vertical team dissemination

Rollout participants will receive the Resource Guide plus a CD containing the 6th and 7th grade CAHSEE standards (to be sent at a later date). Following the rollout, participants will also receive ongoing email support and updates (e.g., future edits/improvements and new ideas).

Target audience:

* Teachers of mathematics (grades 4+), including court school teachers

* Curriculum directors/coordinators; math coaches

* Intervention support personnel

* Site/district/county administrators and staff development leaders

The presenters for each regional rollout will include members of the CAHSEE Mathematics Project leadership team: Patricia Duckhorn, Victor Gee, and Paul Pechin.

Listed below are the names and contact information for the regional leads, along with tentatively scheduled rollout dates:

Region – Contact – County – Phone – Email – Tentative Date


1 – Doreen Heath Lance – Sonoma – (707) 524-2853 – – May 4

1 – Tim Gill – Sonoma – (707) 524-2600 –

2 – Lisa Sandberg – Tehama – (530) 528-7388 – – April 30

2 – Michelle Sanchez – Butte – (530) 532-5806 –

3 – Patricia Duckhorn – Sacramento – (916) 228-2244 – – May 13

3 – Paul Pechin – Sacramento – (916) 228-2631 –

4 – Victor Gee – Alameda – (510) 670-4538 – – May 24

4 – April Cherrington – San Mateo – (650) 802-5359 –

5 – Kirsten Sarginger – Santa Clara – (408) 453-4351 – – May 7

5 – Sandra Ruehlow – Santa Clara – (408) 453 6692 –

6 – Chris King/Diane Waterman – Stanislaus – (209) 525-6602 – – May 6

6 – Garry Potten – San Joaquin – (209) 468-9177 –

7 – Julie Moshier – Tulare – (559) 651-3641 – – May 8

7 – Lori Hamada – Fresno – (559) 497-3729 –

8 – Sheri Willebrand – Santa Barbara – (805) 964-4710 x5267 – – May 2

8 – Kate Dubost – San Luis Obispo – (805) 782-7228 –

9 – Marsha King – Imperial – (760) 312-6464 – – May 16

9 – Cathy Pierce – San Diego – (858) 569-5411 –

10 – Dolores Jones – San Bernadino – (909) 386-2623 – – May 21

10 – Annette Kitagawa – Riverside – (909) 826-6408 –

11 – Cheryl Avalos – Los Angeles – (562) 922-6808 – – May 22

11 – David Moorhouse – Los Angeles – (562) 922-6319 –

(2) California Teachers Selected as Recipients of 2001 Presidential Mathematics and Science Teaching Awards (Press Release: 28 March 2002)

Contact:  Doug Stone or Pam Slater, California Department of Education: 916-657-3027


Four California teachers are recipients of the prestigious 2001 Presidential Awards for Excellence in Mathematics and Science Teaching, announced State Superintendent of Public Instruction Delaine Eastin.

The awardees in science are: Elementary–Julie Taylor, Adelanto Elementary School District, San Bernardino County, and Secondary–Pam Miller, Monterey Peninsula Unified School District, Monterey County. The mathematics awardees are: Elementary–Leanna Baker, Hayward Unified School District, Alameda County, and Secondary–Chris Shore, Temecula Valley Unified School District, Riverside County…

The White House program, established in 1983 and administered by the National Science Foundation, bestows the nation’s highest honor for mathematics and science teachers for kindergarten through grade twelve. The Presidential awardees-who represent every state, the District of Columbia, U.S. Territories, and the U.S. Department of Defense schools-were selected from more than 600 nominees.

After an initial selection process at the state or territorial level, a national panel of distinguished scientists, mathematicians, and educators recommends teachers to receive a Presidential award-one elementary and one secondary mathematics teacher, and one elementary and one secondary science teacher from each state and jurisdiction. Each awardee receives a $7,500 grant to their school and a trip to Washington, D.C. to attend the awards ceremony and participate in other activities…

For more information, see the Presidential Awards for Excellence in Mathematics and Science Teaching’s Web site at You may also contact Charlotte Keuscher-Barkman in the California Department of Education’s Awards Unit at (916) 657-4413.

(3) Scorers Needed for the National Board for Professional Teaching Standards™

Source: Misty Sato –

The National Board for Professional Teaching Standards™ (NBPTS) is seeking teachers to score portfolio or assessment center exercises this summer.  The scoring site in the Santa Clara area will include the following certificate areas:

* Early Adolescence/Mathematics (ages 11-15) —–> July 22-August 2, 2002

* Early Adolescence/English Language Arts (ages 11-15) —–> July 15-26, 2002

* Adolescence and Young Adulthood/English Language Arts (14-18+) —–> July 15-26, 2002

Scorers undergo rigorous training in preparation for scoring candidate responses.  Assessors will critically review performances from candidates from all over the U.S., looking at classroom practices, examples of student work, and essential content knowledge required of accomplished teachers.  Rarely do teachers have such an opportunity to observe other teachers in the performance of their craft.  NBPTS assessors will read responses from at least 100 candidates during a scoring session.  They interact with their peers and have the opportunity to reflect on the requirements for accomplished practice.  Assessors may also apply for graduate credit.  This is also a beneficial experience for teachers wishing to become a candidate for National Board Certification.

Honorarium:  $125.00 per day

Bonus:  $300.00 candidate fee waiver when applying for National Board Certification® in 2002-2003

Qualifications: Baccalaureate degree; valid teaching license/certificate; currently be teaching in the certificate area at least half-time or be a National Board Certified Teacher; three years of teaching experience in a pre-K-12 classroom; not be a current or non-achieving candidate for National Board Certification; successful completion of assessor training (provided by NBPTS)

Scoring sessions will begin at 8:30 am and end at 5:00 pm, Monday-Friday, although Saturday scoring may be required. Lunch will be provided. NBPTS cannot reimburse assessors for travel or lodging expenses.

If you have any questions about the National Board or want to receive more information about assessing this summer, write to (Attn: Dan Hagy) or call 1-800-22TEACH (1-800-228-3224) ext. 8464.



(1) Eighth Annual World’s Largest Math Event: “Entertaining Mathematics”

Source:  National Council of Teachers of Mathematics


The eighth annual World’s Largest Math Event (WLME 8) is slated for April 26, 2002, with the theme “Entertaining Mathematics.”

WLME 8 continues the spirit of celebrating mathematics as an important part of life itself, inviting teachers and students in grades K-16 from around the globe to participate. This event features mathematical investigations and problems related to the entertainment industry. The 12 activities are divided into four sections: Music, Film and Theatre, Print Media, and Television.

This year, the WLME activities are located exclusively on the NCTM Web site. Share these activities with other teachers and interested community members. The teacher’s notes and solutions follow each activity, and a bibliography provides additional resources…

(2) “Looking Out for No. 1–Math Theory Suggests Looking for 1s Can Help Detect Cooked Books” by John Allen Paulos

Source: – 1 March 2002


Was there any way of looking at Enron’s books and–not knowing anything about the company’s specific accounting practices–determining whether the books had been cooked? There may have been, and the mathematical principle involved is easily stated, but counterintuitive.

Benford’s Law states that in a wide variety of circumstances numbers as diverse as the drainage areas of rivers, physical properties of chemicals, populations of small towns, figures in a newspaper or magazine, and the half-lives of radioactive atoms begin disproportionately with the digit “1.”

Specifically, they begin with “1” about 30 percent of the time, with “2” about 18 percent of the time, with “3” about 12.5 percent of the time, and with larger digits progressively less often. Less than 5 percent of the numbers in these circumstances begin with the digit “9.”

(This is in stark contrast to many other situations–say where a computer picks a number between 0 and 100 at random–where each of the digits from “1” to “9” has an equal chance of appearing as the first digit.)…

A mathematically inclined accountant, Mark Nigrini, generated considerable buzz when he noted that Benford’s Law could be used to catch fraud in income tax returns and other accounting documents.

The following example suggests why collections of numbers governed by Benford’s Law arise so frequently:

Imagine that you deposit $1,000 in a bank at 10 percent compound interest per year. Next year you’ll have $1,100, the year after that $1,210, then $1,331, and so on. The first digit of your bank account remains a “1” for a long time.

When your account grows to more than $2,000, the first digit will remain a “2” for a shorter period as your interest increases. And when your deposit finally grows to more than $9,000, the 10 percent growth will result in more than $10,000 in your account the following year and a long return to “1” as the first digit.

If you record the amount in your account each year for a large number of years, these numbers will thus obey Benford’s Law.

The law is also “scale-invariant” in that the dimensions of the numbers don’t matter. If you expressed your $1,000 in euros or francs or drachmas and watched it grow at 10 percent per year, about 30 percent of the yearly values would begin with a “1,” about 18 percent with a “2,” and so on.

More generally, Hill showed that such collections of numbers arise whenever we have what he calls a “distribution of distributions,” a random collection of random samples of data. Big, motley collections of numbers follow Benford’s Law.

And this brings us back to Enron, accounting, and Nigrini, who reasoned that the numbers on accounting forms, which come from a variety of company operations, each from a variety of sources, fit the bill and should be governed by Benford’s Law.

That is, these numbers should begin disproportionately with the digit “1,” and progressively less often with bigger digits, and if they don’t, that is a sign that the books have been cooked. When people fake plausible-seeming numbers, they generally use more “5s” and “6s” as initial digits, for example, than would be predicted by Benford’s Law.

Nigrini’s work has been well-publicized and has no doubt been noted by accountants and by prosecutors. Whether the Enron and Arthur Andersen people have heard of it is unclear, but investigators might want to check if the percentage of leading digits in the Enron documents is what Benford’s Law predicts. Such checks are not fool-proof and sometimes lead to false positive results, but they provide an extra tool that might be useful in certain situations.

It would be amusing if, in looking out for No. 1, the culprits forgot to look out for their “1s.” Imagine the Andersen shredders muttering that there weren’t enough leading “1s” on the Enron documents they were feeding into the machines. A 1-derful fantasy!


Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on appears every month. [For an archive of his Who’s Counting? columns, see

(3) “The Secret Lives of Numbers”


The authors conducted an exhaustive empirical study with the aid of custom software, public search engines and powerful statistical techniques, in order to determine the relative popularity of every integer between 0 and one million. The resulting information exhibits an extraordinary variety of patterns which reflect and refract our culture, our minds, and our bodies.

For example, certain numbers, such as 212, 486, 911, 1040, 1492, 1776, 68040, or 90210, occur more frequently than their neighbors because they are used to denominate the phone numbers, tax forms, computer chips, famous dates, or television programs that figure prominently in our culture. Regular periodicities in the data, located at multiples and powers of ten, mirror our cognitive preference for round numbers in our biologically-driven base-10 numbering system. Certain numbers, such as 12345 or 8888, appear to be more popular simply because they are easier to remember.

Humanity’s fascination with numbers is ancient and complex. Our present relationship with numbers reveals both a highly developed tool and a highly developed user, working together to measure, create, and predict both ourselves and the world around us. But like every symbiotic couple, the tool we would like to believe is separate from us (and thus objective) is actually an intricate reflection of our thoughts, interests, and capabilities. One intriguing result of this symbiosis is that the numeric system we use to describe patterns, is actually used in a patterned fashion to describe.

We surmise that our dataset is a numeric snaphot of the collective consciousness. Herein we return our analyses to the public in the form of an interactive visualization, whose aim is to provoke awareness of one’s own numeric manifestations.


The Secret Life of Numbers by Golan Levin, et. al (February 2002) is a commission of New Radio and Performing Arts, Inc., for its Turbulence web site. It was made possible with funding from The Greenwall Foundation.

(4) “Students May Be Learning More About Avoidance Strategies Than Arithmetic In Math Class; Study Shows What Teachers Can Do” 

Article’s lead author:  Julianne C. Turner: (574) 631-3429;


“Please don’t call on me” can be a pervasive thought by students who are not doing well in math class. By early adolescence, it is common for some students to become experts in avoidance strategies — avoiding asking for help when they need it, withdrawing effort and resisting novel approaches to learning — in order to deflect attention from low ability. This type of behavior can cause students to fall further behind academically and may eventually lead some to drop out of school. But new research shows that teachers that emphasize learning rather than performance may help prevent this self-destructive behavior. The findings appear in the current issue of the Journal of Educational Psychology, published by the American Psychological Association (APA).

In their study involving 1,092 sixth-grade students in 65 sixth-grade classrooms in four ethnically and economically diverse school districts in three Midwestern states, Julianne C. Turner, Ph.D. of the University of Notre Dame and co-authors surveyed the students to determine use of avoidance strategies. The study also included the use of trained observers who watched and audiotaped nine of the students’ teachers while they taught their math classes.

The researchers found that students reported using fewer avoidance techniques in classrooms perceived as emphasizing learning, understanding, effort and enjoyment. In those classrooms, teachers helped students who had problems understanding, gave them opportunities to demonstrate new competencies and provided substantial motivational support for learning. Teachers in these “mastery-oriented” classrooms made sure their students did not feel inadequate or ashamed when they did not understand. “By modeling their own thinking processes, these teachers demonstrated that being unsure, learning from mistakes, and asking questions were natural and necessary parts of learning,” according to the authors. By contrast, “students reported higher incidences of avoidance strategies in classrooms in which teachers devoted little attention to helping students build understanding and in which motivational support was low.”

The teacher observations, say the authors, provided valuable insight into how teaching methods affect avoidance behaviors. For instance, in a classroom where students used more avoidance strategies, the teacher placed greater emphasis on getting an answer correct, with little discussion about the important concepts in a lesson and little explanation of why an answer was correct. If a student did not know the answer, the teacher would ask another student and did not usually stop to explain the answer. “Because the teacher typically did not respond to mistakes and misunderstandings with explanations or allow students to explain their strategies, his students may have felt vulnerable to public displays of incompetence and adopted more avoidance strategies,” explained the researchers.

In classrooms where students used fewer avoidance strategies, the teachers tended to model, hint and elicit support from other students to help their students learn. In those classrooms, the students were active participants in instructional discourse that stressed understanding and explanation. “Perhaps because they knew their teachers and peers would help, students in these classrooms did not seem to need to adopt avoidance strategies to appear able to others,” said the authors.

Classrooms with students who reported using avoidance strategies less also had teachers that used math-related humor as part of their lessons. Humor may lessen tension and encourage students to view their math classes as more enjoyable, say the authors.

Citation “The Classroom Environment and Students’ Reports of Avoidance Strategies in Mathematics: A Multimethod Study,” Julianne C. Turner, University of Notre Dame, Carol Midgley, University of Michigan, Debra K. Meyer, Elmhurst College, Margaret Gheen, University of Michigan, Eric M. Anderman, University of Kentucky, and Yongjin Kang, University of Michigan; Journal of Educational Psychology, Vol. 94, No. 1.