Where’s the Math? – DEBRA J. SAUNDERS


Where’s the Math?
DEBRA J. SAUNDERS
Sunday, October 17, 1999

 

THIS MONTH, the U.S. Department of Education came out with a list of 10 “exemplary” or “promising” math education programs. Kings County fourth-grade teacher Doug Swords was shocked at the department’s bad choices.

Some three years ago, his school district adopted MathLand, a math curriculum that prefers not to give lessons with “predetermined numerical results.” The department of Educrats, oops, I mean, Education, rated MathLand as “promising.” Today, he said 14 out of 18 teachers use MathLand only as a supplement. “I stashed away my Addison-Wesley textbooks, as did a few other teachers,” he explained.

Do you teach your students how to multiply? I asked him. (You wouldn’t think that would be something I’d have to ask, but these days, it is.) Yes, he said. Is MathLand helpful in teaching kids to multiply? “No, quite frankly,” Swords answered.

UC Berkeley math professor Hung-Hsi Wu couldn’t believe the department described MathLand as “promising.” He’d describe MathLand as “execrable.”

Or how about: “I can’t believe it’s math class.” A second-grade MathLand exercise called Fantasy Lunch instructs students to think up their fantasy lunch, draw it on paper, then cut out the “food” and place their drawings into a bag.

A frantic teacher wrote to me two years ago, furious that she had spent 75 minutes on that exercise and there was no math in it. It was “like therapy,” she said. On more than one occasion, her students asked her, “Can we do some real math now?”

Wu had problems with the other nine picks as well. While there were things he liked about the high school programs, they lacked what he called “mathematical closure. You start something, you ought to finish it.”

He said almost all of his students took more traditional math classes not cited as “exemplary” or “promising” by the Department of Education. That wouldn’t surprise Melissa Lynn, who got As in high-school math, then placed in the bottom 1 percent in the University of Michigan math placement test. She blames the Core-Plus program which the department rated as “exemplary.” “It had very good intentions, and wanted you to apply real principles to real life scenarios,” she explained this spring, “but it was missing the crucial element of algebra.“

Wayne Bishop, a math professor at Cal State L.A. who is the Ralph Nader of math curricula, sees the department’s move as a reaction against California’s return to math sanity — after a mad fling when state educrats embraced “there is no right answer” new-new math curricula.

He’s right. The selection panel appoint ed by the department had as a main criterion that the math series ascribe to trendy standards put out by the National Council for Teachers of Mathematics (NCTM).

Don’t ask me why. Last year Bishop looked at the scores of some of the students subjected to the brilliance of new- new math wizards. In 1995, NCTM Chairman Jack Price boasted about a program on which he worked. Turns out, Price’s star school ranked in the bottom quartile nationally in the STAR test last year. Only 12 percent of the school’s eighth graders scored above the national average. Price called that a successful program.

The department cited data that show schools whose test scores improved with MathLand. Bishop isn’t impressed. “They appear to have excluded data where MathLand scores dropped,” he noted.

An administrator from an urban district that stopped using MathLand had just visited a school that had seen a 27 percent increase in its math scores after buying a traditional math series that didn’t rate in the department’s Top 10. Under ideal circumstances, he said, MathLand could work, but urban districts don’t have too many ideal circumstances.

Bill Evers of Stanford’s Hoover Institution called the department’s Top-10 picks “unconditional surrender to fuzziness.”

Fuzziness? The department praised one K-6 math program because, “Features include problem solving; linking past experience to new concepts; sharing ideas; developing concept readiness through hands-on explorations; cooperative learning through small-group activities; and home-school partnerships.”

Sounds more like marriage counseling than math class.

The problem: It’s not the kids who need counseling here. It’s the adults who care so little about children’s success that they would assert that Fantasy Lunch makes for a “promising” math program.

 

You can reach Debra J. Saunders on The Gate at sfgate.com.


©2005 San Francisco Chronicle

Was the Texas State Board of Education correct to reject the 3rd Grade curriculum of Everyday Math?

Was the Texas State Board of Education correct to reject the 3rd Grade curriculum of Everyday Math?

By Nakonia (Niki) Hayes
Columnist EdNews.org

Part One: The facts on Texas Mathematics Standard 3.4(a) and Everyday Math
Part Two: The facts on the physical construction of Everyday Math, 3rd Grade, 3rd Edition
Part Three: An introduction to the author of this report

Faulty construction was the complaint of the Texas State Board of Education (SBOE) of Everyday Math‘s third grade materials up for adoption in November of this year. It wasn’t about the 109, 263 proofreading mistakes in mathematics materials submitted by all the publishers this year in Texas. It was about an issue that speaks to the heart among many in the mathematics education debate today: the multiplication tables.

With a 7-6 vote (and one abstention), the SBOE rejected the third grade program of the third edition of Everyday Math. They said it did not meet the Texas mathematics standards that required third graders to learn their multiplication tables to “automaticity” (or “by heart”) through the 12s:

3.4(a) The student is expected to learn and apply multiplication facts through 12 by 12 using concrete models and objects.

The SBOE’s members who voted to reject the book have been blasted by Everyday Math representatives who maintain the book does meet the standard. They say with proofreading corrections the problem can be corrected. There’s also a fringe group who declares the board is “censoring” materials by their voting to reject the 3rd grade curriculum. Lawsuits are being threatened.

Let’s look at the facts. In their Math Masters and Home Link Masters, which is composed of worksheets to be sent home to the family, Everyday Math states in the “Introduction of Unit 7, pp. 202-232, on “Multiplication and Division”:

“The goal is for children to demonstrate automaticity with x0, x1, x2, x5, and x10 multiplication facts and to use strategies to compute remaining facts up to 10 x 10 by the end of the year.”

Conclusion: “Automaticity” is not expected for x3, x4, x6, x7, x8, x9, and certainly not for x11 and x12.

Yet, in their Correlation to the Texas Knowledge and Skills (TEKS) manual, EM lists 39 pages that say the publisher has lessons/activities to support the Texas standardrequirement of learning and applying multiplication facts through 12 by 12.

All of those pages do support multiplication topics. Only one page, however, refers to working with 12 by 12 multiplication facts: P. 280 in the Teacher’s Lesson Guide (TLG) Vol. 1 says in a framed graphic at the bottom of the page:

ADJUSTING THE ACTIVITY (of “Playing Baseball Multiplication”): The basic game uses facts through 6 x 6. The advanced version of Baseball Multiplication, described on pages 276-277 in the Student Reference Book, uses products up to 12 x 12. Children use Math Masters, page 444 to keep score.

Since it is a Teacher’s Guide that determines the “objective for the lesson” each day, and with which supplemental workbooks or resources are correlated, it is important to review the objectives cited as “supporting” Texas Math Standard 3.4(a):

1) Vol. 1, P. 248-253: To provide opportunities to use arrays, multiplication/division diagrams, and number models to represent and solve multiplication number stories. (Lesson 4.3, “Multiplication Arrays”)

2) Vol. 1, P. 272-277: To review fact families and the Multiplication/Division Facts Table; and to guide children as they practice multiplication and division facts. (Lesson 4.6, “Multiplication and Division Fact Families”)

3) Vol. 1, P. 278-280: To practice multiplication facts. (Lesson 4.7, “Baseball Multiplication.”)

4) Vol. 2, P. 576-579: To review square-number facts, multiplication, and division patterns. (Lesson 7.1, “Patterns in Products”)

5) Vol. 2, P. 582-587: To guide children as they determine which multiplication facts they still need to learn. (Lesson 7.2, “Multiplication Facts Survey”)

Other sources cited by EM that support Texas Math Standard 3.4(a) include four worksheet activities in the Student Math Journal, four pages of review in the Student Reference Journal, and three activities in Math Masters (not an assessment source).

Two conclusions can be drawn from this factual information. First, Everyday Math does not meet the Texas Mathematics Standards for 3rd Grade. Second, the claim by representatives that it does is, at best, a misinformed one among themselves. At worst, the claim is a deliberate attempt to mislead the board.

The academic value placed by some Texas state board members on the role of automaticity with multiplication facts [i.e., cognitive, not calculator], with 3rd grade students learning and applying required mathematical knowledge and skills for future success, was made clear with their votes.

This mistake by Everyday Math is not a “proofreading” error. It is a clear indicator of the publisher’s philosophy regarding multiplication facts for 3rd graders. It will also require more than a “supplement” to correct or improve the program.

Part Two: The facts on the physical construction of Everyday Math, 3rd Grade, 3rd Edition

As a preface, it is necessary to remind every adult involved (or not) with elementary education the daily schedule of those teachers. They are teaching the four core subjects of mathematics, language arts (reading/writing, which includes phonemic awareness, spelling, and grammar), social studies, and science. Most are required to cover other areas such as art, music, character education, keyboarding skills, etc.

When a 2003 survey was done in Seattle Public Schools’ to determine actual “seat time” for learning purposes, we discovered that elementary students received about 4.5 hours of actual academic learning time per day out of the 6.5 hours we had them for 176 student days. (This excludes holidays or days off for teacher in-service, etc, during the year). The limited availability of learning time had become apparent after we removed the following times of “non-academics” that had to take place each day:

  • “Passing time”: Bringing children in from their lineup areas when the bell rang to start the school day and settling them into class routines.
  • Recess twice daily (once in the morning and once in the afternoon).
  • “Passing time x2”: Bringing children in from each recess and settling them into class routines.
  • “Passing time: Taking children to the cafeteria for lunch (and going to restrooms before lunch

and after recess to wash hands, etc.)

  • “Passing time”: Bringing children in from lunch recess and settling them into class routines.
  • “Passing time”: Taking children to and retrieving them from the gym for P.E.
  • “Passing time”: Taking children to and retrieving them from the library.
  • Setting aside time for foreign language instruction, or, at our school, other electives such as journalism or photography to improve writing skills (which included the 3rd grade).
  • Preparing children for the end of the school day with assignments, coats, etc., and ready for buses.

Then, we subtracted times for early dismissals (teacher professional development, holidays, parent-teacher conference weeks), assemblies, field trips, fire drills, earthquake drills, lockdown drills, two class parties allowed each year, and at least two weeks for TESTING for the Direct Reading Assessment, Direct Writing Assessment, and the state-mandated WASL (Washington Assessment of Student Learning).

There are, of course, other events that cause children to be out of their “academic setting.” One of the biggest is the absence of the regular teacher from the classroom. Substitutes are generally warm bodies who babysit the students. (Add a curriculum with unique and/or unknown procedures, such as lattice multiplication and substitutes will definitely not be able to teach math.) High teacher absenteeism means lost learning time. Another major source of time lost is caused by student misbehaviors that take away from a classroom’s learning environment.

We realized we were lucky to get 4.5 hours per day for “real learning.”

It is therefore crucial to remember the class schedules, hours, and academic requirements that are expected of elementary teachers as the following information is reviewed from Everyday Math‘s 3rd grade curriculum materials.

The number of pages in each published “component” (manual, workbook, journal, etc.) is shown in red font. Worksheets, or “paper-and-pencil” tasks, are shown in blue font. For those who claim they avoid the “traditional” use of paper-and-pencil tasks and thus prefer Everyday Math’s “activity-based learning,” this should be enlightening.

A total of 2,997 pages are in the 3rd grade “components” of Everyday Math for teacher review and use. Of those, 925 are worksheets for students.

In addition, when the following materials are stacked, they measure 8 inches in height and weigh 18 pounds. The following body of information, without the page count, is from http://www.wrightgroup.com/index.php/componentfeatures?isbn=007608972X.

Grade 3 Core Classroom Resource Package includes:

  • Teacher’s Lesson Guides (Volume 1 & 2) – The core of the Everyday Mathematics program, the Teacher’s Lesson Guide provides teachers with easy-to-follow lessons organized by instructional unit, as well as built-in mathematical content support. Lessons include planning and assessment tips and multi-level differentiation strategies to support all learners. 848 pages + 152 pages of “reference” information (glossary, charts, etc.) = 1000 pages.
  • Teacher’s Reference Manual (Grades 1-3) – Contains comprehensive background information about mathematical content and program management. 290 pages
  • Assessment Handbook – Grade-specific handbook provides explanations of key features of assessment in the Everyday Mathematics program. Includes Assessment Masters. 144 pages of examples for teacher information + 83 forms for teacher use = 227 pages + 64pages of blank assessment worksheets
  • Differentiation Handbook – Grade-specific handbook provides that helps teachers plan strategically in order to reach the needs of diverse learners. 145pages
  • Home Connection Handbook (Grades 1-3) – Enhances home-school communication for teachers and administrators. Includes masters for easy planning. 102 pages
  • Minute Math (Grades 1-3) – Contains brief activities for transition times and for spare moments throughout the day. 112 pages
  • Math Masters – Blackline masters for routines, activities, projects, Home Links/Study Links, and games. 468pages
  • Number Grid Poster
  • Sunrise/Sunset Chart
  • Content by Strand Poster
  • One set of Student Materials
    • Student Math Journals (Volumes 1 & 2) – These consumable books provide daily support for classroom instruction. They provide a long-term record of each student’s mathematical development. 281 pages
    • Student Reference Book (Grades 3) – This book contains explanations of key mathematical content, along with directions to the Everyday Mathematics games. 308 pages
    • Pattern Block Template – A clear, green, plastic tracing template contains a variety of geometric shapes with six of the shapes exactly matching the sizes of the pattern blocks.

To cover the Texas Mathematics Standards, the two volumes of the Teacher’s Lesson Guide would have to be completed; i.e., 848 pages of teaching directions and content in 176 days, supported by assignments from other EM “component” materials. Even though each day’s lesson/objective in the Guide covers an average of four pages, that still requires 212 days to get through all of the recommended lessons—without assessments. And that’s assuming each day’s lesson is accomplished in one day, with no “extensions” or reteaching required of the lesson.

A buzz phrase now being circulated by Everyday Math‘s publisher, the University of Chicago Center for Elementary Mathematics and Science Education, is “fidelity of implementation” (FOI), explained at http://cemse.uchicago.edu/node/3. This leads one to believe that only EM materials may be used to assure promised outcomes by the publisher, which also suggests a district should buy all of the above materials or EM should not be held accountable for negative learning results.

Questions come to mind

The first question has to center on the costs, in millions of dollars, for these materials. That includes the published materials and the professional development required to train teachers how to use those “effectively”—or at least with “fidelity.” And, in today’s “eco-friendly world,” the costs to the environment should be considered with how many trees it takes to print these materials.

Second, since such massive teaching resources seem to be written for every conceivable situation a teacher might face in a 3rd grade mathematics class, what is the underlying message? Is this an effort to “teacher-proof” the materials?

Third, has any data been collected to see how much of EM material is actually covered each year?

Lastly, has anyone surveyed third grade teachers to see if they need or want 3,000 pages of materials to cull for only ONE of the four core subjects they must teach in 176 days?

Part Three: An introduction to the author, Nakonia (Niki) Hayes

Even though I retired from teaching in 2006, my interest in mathematics education has continued. For that reason, I decided to study the issues around the recent rejection of Everyday Math‘s third grade curriculum by the Texas State Board of Education. In order to understand my frame of reference in reporting on this situation, I offer a brief introduction to my background as an educator and journalist.

First, my bachelor’s degree is in journalism; my master’s, in counseling. I began my doctoral work in mathematics education at the University of Texas-Austin but decided its philosophy did not square with mine and I left the program. My work in journalism fields for 17 years included being a newspaper reporter, public information officer, and in public relations positions for two state senators. I’ve taught journalism in three high schools, at a community college, and I established a journalism program at a K-5 elementary school in 2001 that is still being used.

As an educator for 28 years, I became certified and experienced in special education, counseling, mathematics, and administration. While working as a special education teacher, I found that teaching mathematics to my middle and high school learning disabled students was a valuable way for them to learn structure, cause-and-effect, and linear thinking, all traits they needed to incorporate in their episodic learning and living. This led me to earn a certification in mathematics. Subsequently, I taught grades 6-12 for 15 years in high, at-risk populations in Central Texas and Washington state. My students were in special, regular, and gifted education classes. Because of my training, I was usually asked to take the English language learners.

I became acutely aware of the growing deficiencies of math skills among all of my students during 1987-1991. My question was, “What is happening at the elementary level that our students are coming to us with so many deficiencies in mathematics?”

As a middle and high school guidance counselor, I saw failure rates and “remedial courses” becoming the norm for students, both in public education and colleges.

I figured being a principal would allow me to affect curriculum and the teaching of mathematics in elementary schools. So, I became one. That experience includes my being a P-12 principal/ teacher on an American Indian reservation and principal at a K-5 school in Seattle, WA, with an 80%, upper-middle-class white, student population.

Lastly, my training in Jerusalem, Israel in 1998 and 1999 with Reuven Feuerstein introduced me to a pioneer in constructivism who knows how to use it effectively. Prof. Feuerstein’s International Center for the Enhancement of Learning Potential is dedicated to teaching cognitive remediation strategies. His work has been used throughout Europe with business leaders, in South Africa for children from apartheid policies, and for a half million children in Brazil. The American “home office” is with IRI Skylight Publishers in Chicago, IL, since there are some U.S. districts that use his programs.

In essence, my approach to mathematics education is not that of a mathematician. It is one of a “diversified” educator who happens to appreciate the reality and potential of mathematics and what it can mean for learners who master its power.

And in summary, this report isn’t about pedagogy, the primary focus of the math wars across this nation. It is about the accuracy of EM‘s content as compared to the Texas state standards and an accountability of simple quantitative facts. As a teacher and an administrator, I would not accept any program that clearly maintains the mile-wide-inch-deep approach in mathematics education, whether it’s in teacher materials or in content for students, and that is what Everyday Math offers.

Published November 27, 2007

Understanding the Revised NCTM Standards Arithmetic is Still Missing!

Understanding the Revised NCTM Standards

Arithmetic is Still Missing!

by Bill Quirk  ( wgquirk@wgquirk.com)

Contrary to Recent Reports, the NCTM Has Not Changed Its Philosophy

On April 12, 2000, The National Council of Teachers of Mathematics (NCTM)  released Principles and Standards for School Mathematics (PSSM), a 402 page revision of  the NCTM Standards. The next day The New York Times reported: “In an important about-face, the nation’s most influential group of mathematics teachers announced yesterday that it was recommending, in essence, that arithmetic be put back into mathematics, urging teachers to emphasize the fundamentals of computation rather than focus on concepts and reasoning.”  It was further reported that “the council added strong language to its groundbreaking 1989 standards, emphasizing accuracy, efficiency and basic skills like memorizing the multiplication tables.”

Compare the preceding New York Times quotes to the following contradictory quote, published by the NCTM (in the third PDF file, Commonsense Facts to Clear the Air) under “News and Hot Topics” at  NCTM Speaks Out: Setting the Record Straight about changes in Mathematics Education.

When calculators can do multidigit long division in a microsecond, graph complicated functions at the push of a button, and instantaneously calculate derivatives and integrals, serious questions arise about what is important in the mathematics curriculum and what it means to learn mathematics. More than ever, mathematics must include the mastery of concepts instead of mere memorization and the following of procedures. More than ever, school mathematics must include an understanding of how to use technology to arrive meaningfully at solutions to problems instead of endless attention to increasingly outdated computational tedium.   -NCTM,  Commonsense Facts to Clear the Air

It’s clear that The New York Times was fed misleading NCTM propaganda, perhaps designed to placate “math wars” opponents.  Not surprisingly, we will show here that the NCTM has not rediscovered arithmetic.  Similar to the original NCTM Standards, PSSM is vague about the major components of arithmetic mastery:

    1. Memorization of of basic number facts
    2. Mastery of the standard algorithms of multidigit computation.
    3. Mastery of fractions

The NCTM has toned down the constructivist language, but they still stress content-independent “process skills” and student-centered “discovery learning”.  Similar to the NCTM Standards, PSSM emphasizes manipulatives, calculator skills, student-invented methods, and simple-case methods.

Although PSSM contains five “Connections” sections,  there continues to be no acknowledgement of the vertically-structured nature of  mathematics.  Mastery of math requires a step-by-step build up (in the brain) of specific content knowledge.  PSSM omits this aspect of the “connections” within mathematics.  The idea conveyed by the following example is not found in PSSM.

Example: Migrating up the math learning curve

Each of the following skills serves as a preskill for acquiring all higher skills. To move up to the next skill level, the student must remember all preskills.

  1. The ability to instantly recall basic multiplication facts
  2. The ability to factor integers
  3. The ability to reduce a fraction to lowest terms.

The NCTM says they want to maximize “understanding”, but they still fail to recognize that specific math content must first be stored in the brain as a necessary precondition for understanding to occur.  Although rarely the preferred method, intentional memorization is sometimes the most efficient approach.  The first objective is to get it into the brain!  Then newly remembered math knowledge can be connected to previously remembered math knowledge and understanding becomes possible. You have to “know math” before you can “understand math”, “do math”, or “solve math problems.”

We conclude this introductory section by noting that there is evidence of a battle within the NCTM, with some voices crying out for genuine arithmetic. These voices were heard in the Principles and Standards for School Mathematics: Discussion Draft (PSSM Draft), published in October, 1998. At later points in this document you will find quotes from both PSSM and PSSM Draft. The quotes from PSSM Draft do not appear in PSSM, the final version published in April, 2000.  The voices of reason have been largely silenced!  Here is an example of the silencing.

Technology: PSSM Draft vs. PSSM

However, access to calculators does not replace the need for students to learn and become fluent with basic arithmetic facts, to develop efficient and accurate ways to solve multidigit arithmetic problems, and to perform algebraic manipulations such as solving linear equations and simplifying expressions.  -PSSM Draft, Page 43

Technology should not be used as a replacement for basic understandings and intuitions; rather it should be used to foster those understandings and intuitions.  -PSSM, page 24

They Still Call It “Arithmetic”

Mastery of Basic Facts or Derive Them When Needed?

Similar to the original NCTM Standards, PSSM fails to clearly acknowledge that the ability to instantly recall basic number facts is an essential preskill, necessary to free up the mind, first for mastery of the standard algorithms of multidigit computation, and next for mastery of fractions. Then, once this knowledge is also instantly available in memory, the mind is again free to focus on the next level, algebra.

As the quotes below show, the PSSM Draft emphasized quick recall, but it appears that PSSM has reverted to the NCTM Standards idea that “the ability to efficiently derive” is preferable to “the ability to instantly recall.”

Quotes From The PSSM Draft:  Recall Number Facts Quickly

Most students should be able to recall addition and subtraction facts quickly by the end of grade 2 and recall multiplication and division facts with ease and facility by the end of grade 4.  -PSSM Draft, Page 51

A certain amount of practice is necessary to develop fluency with both basic fact recall and computation strategies for multi-digit numbers.  Anderson, Reder, and Simon (1996) point out that practice is clearly essential for acquiring cognitive skills of almost any kind. -PSSM Draft, Page 114

Quotes From PSSM:  Derive Number Facts Quickly

Fluency with basic addition and subtraction number combinations is a goal for pre-K-2 years.  By fluency we mean that students are able to compute efficiently and accurately with single-digit numbers.   -PSSM, Page 84

When students leave grade 5, they should be able to . . . efficiently recall or derive the basic number combinations for each operation.  -PSSM, Page 149

Fluency with the basic number combinations develops from well understood meaning for the four operations and from a focus on thinking strategies. -PSSM, Page 152-153

If by the end of the fourth grade, students are not able to use multiplication and division strategies efficiently, then they must either develop strategies so that they are fluent with these combinations or memorize the remaining “harder” combinations. -PSSM, Page 153

The changes may be subtle, but notice that fluency with basic number facts is defined as the ability to “compute efficiently and accurately.”   It does not mean the ability to instantly recall.  Also, in PSSM it’s “recall or derive” by the end of grade 5, not “recall” by the end of grade 4 as recommended in the PSSM Draft.  Basic number fact “thinking strategies” appear to be preferred by the writers of PSSM.  They give grudging admission that memorization may be necessary.  There is some discussion of activities to teach fact relationships, but there is no discussion of mastery activities to facilitate fact memorization.

Mastery of Standard Algorithms or Student-Invented Algorithms?

Considering that the NCTM appears to prefer basic number fact derivation strategies, it’s not surprising that they also appear to prefer student-invented algorithms for multidigit computation. Here are some  relevant quotes from PSSM:

In the past, common school practice has been to present a single algorithm for each operation.  However, more than one efficient and accurate computational algorithm exists for each arithmetic operation.   In addition, if given the opportunity, students naturally invent methods to compute that make sense to them.  -PSSM, Page 153

Many students are likely to develop and use methods that are not the same as the conventional algorithms (those widely taught in the United States).  For example, many students and adults use multiplication to solve division problems or add starting with the largest place rather than with the smallest.  The conventional algorithms for multiplication and division should be investigated in grades 3 – 5 as one efficient way to calculate. -PSSM, Page 155   (bold emphasis added)

Students’ understanding of computation can be enhanced by developing their own methods and sharing them, explaining why their methods work and are reasonable to use, and then comparing their methods with the algorithms traditionally taught in school.  In this way, students can appreciate the power and efficiency of the traditional algorithms and also connect them to student-invented methods that may sometimes be less powerful or efficient but are often easier to understand.  -PSSM, Page 220

This last quote contains a hint of truth.  If a student is given a problem and allowed to struggle for a while, trying to solve the problem, then the student becomes motivated to listen and learn about the most efficient, general solution to the problem.  This is the essence of the lesson method used in Japan.  The mistake is to elevate the value of “easier to understand” student-invented methods, while not stressing the power, mathematical importance, and universal acceptance of the efficient, general “algorithms traditionally taught in school.”

Efficient, Accurate, and (Possibly) General Methods

On page 32 of PSSM,  the term “computational fluency” is defined as “having and using efficient and accurate methods for computing”.  Later on the same page, we are told that students should “see the usefulness of methods that are efficient, accurate, and general.”  On page 87  we are told  “Teachers also must decide what new tasks will challenge students and encourage them to construct strategies that are efficient and accurate and that can be generalized.”

Has the definition of computational fluency been (appropriately) expanded to include “general”?   No, the original definition of computational fluency, including only efficient and accurate,  is restated on pages 79 and 153.   It appears that some PSSM writers recognized that all three characteristics contribute to the power of the standard algorithms of arithmetic.  But the standard algorithms are not mentioned in this context (page 87).  Instead, on this page we are advised:  “As students encounter problem situations in which computations are more cumbersome or tedious, they should be encouraged to use calculators to aid in problem solving.”

The Truth is in The Examples

Within the five “Number and Operations” sections, PSSM includes (only) three illustrations of  multidigit computation. All are “student-invented” strategies.  On page 85 we learn “In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different.”  There is no discussion of the accuracy, efficiency or generality of any method found in these three illustrations. Apparently those who wrote about “computational fluency” failed to communicate with those who developed the examples.

  1. Figure 4.3 on page 85 presents six student solutions for computing 25 + 37. Student 2’s method utilized 12 “tallies” (four vertical marks crossed by a horizontal mark) followed by two additional vertical marks, with the 12 tallies identified by  5 written above the first tally, 10 written above the second tally, 15 written above the third tally, and so on until 62 is written above the concluding pair of vertical lines.  The NCTM is pleased with the “completeness” of Student 2’s thinking.  Student 4 correctly used the standard algorithm for addition, but the NCTM appears not to notice, even remarking that student 4’s thinking is “not as apparent.”
  2. Figures 4.4 and 4.5 on page 86 describe student strategies for computing 153 + 273. Randy’s method is described first.  He used beans, bean sticks (10 beans), and rafts of bean sticks (100 beans).  The “conventional algorithm” is used successfully by some nameless students,  but unsuccessfully by other nameless students.  “Becky finds the answer using mental computation and writes nothing down except her answer.”    Subtle, but effective.  Randy and Becky are worth recognizing by name.
  3. Page 153 presents two student solutions for dividing 728 by 34.  Henry used the method of repeated subtraction of multiples of 10, which he apparently invented. Michaela used long division, which she apparently invented. Mrs. Sparks “saw the relationship between the two methods described by the students, but she doubted that any of her students would initially see these relationships”.  This is a surprising lack of confidence, considering the remarkable discovery abilities demonstrated by Henry and Michaela.(We used “long division” to briefly describe the method used by Michaela, but the phrase long division in not found in PSSM.)

In this last illustration, we again we have a hint of the lesson method used in Japan.  But we learn that Mrs. Sparks objective is to “help the students understand, explain, and justify their computational strategies,”  rather than working to achieve closure by  connecting the two methods and teaching long division, emphasizing the efficiency and generality of the long division algorithm.

Mastery of Fractions or Simple-Case Methods for “Familiar Fractions”?

Consistent with the lukewarm treatment of number fact recall, PSSM fails to emphasize the importance of the ability to factor integers, and PSSM never discusses any of the details related to the addition, subtraction, multiplication, division, and simplification of fractions.  The phrase “common denominator” is not found in PSSM.

The NCTM says they want students to “develop and analyze algorithms for computing with fractions” and “develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.”  – PSSM, Page 214.

PSSM’s treatment of fractions offers just two illustrations, one for comparing fractions,  and the other for dividing fractions.  Both are simple-case methods, and neither is efficient or general.

  • For comparing 7/8 to 2/3,  PSSM recommends the use of physical “fraction strips”, never mentioning the concept of converting to a common denominator.  -PSSM, Page 216
  • For dividing 5 by 3/4 they recommend the method of “repeated subtraction,” after first suggesting (see the following quote) that  “invert and multiply” is too difficult for today’s kids.
    • How about 3/4 divided by 5 using repeated subtraction?  Do they expect that  kids will find it easy to use repeated subtraction to show that 9/11 divided by 3/121 equals 33?  No, they will tell you that these are unreasonable divisions, and 11 and 121 are unreasonable denominators (see Connecticut).  They say that students “need to see and explore a variety of models of fractions, focusing primarily on familiar fractions such as halves, thirds, fourths, fifths, sixths, eighths, and tenths.”  -PSSM, Page 150
    • A comment from Professor Richard Askey:  “Given that the authors had a very nice chapter on this topic in Liping Ma’s book with varied word problems and comments  from teachers about such things as objecting to using 1 3/4 divided by 1/2 to see if students understood division of fractions since this is so easy to do without understanding how to divide fractions, I find it shocking that successive subtraction is pushed as the way to do division of fractions, and the final step when successive subtraction does not work is just 1/4 divided by 1/2.  Even that is not adequately explained, it is just done.  As the Chinese teacher suggested, this is too
      easy to see if division of fractions is understood or not.”

      • Note: For the 5 divided by 3/4 problem discussed above in PSSM, the final step, 1/2 divided by 3/4, is not explained.  They just say that 2/3 is left after 6 subtractions of 3/4.
      • How does the student actually carry out the 6 subtractions of 3/4?  We are told “students can visualize repeatedly cutting off 3/4 yard of ribbon” from 5 yards of ribbon.  One wonders if they use scissors to help them “visualize”.

PSSM on “invert and multiply””

The division of fractions has traditionally been quite vexing for students.  Although “invert and multiply” has been a staple of conventional mathematics instruction and although it seems to be a simple way to remember how to divide fractions, students have for a long time had difficulty doing so.  Some students forgot which number is to be inverted, and others are confused about when it is appropriate to apply the procedure.  A common way of formally justifying the “invert and multiply” procedure is to use sophisticated arguments involving the manipulation of algebraic rational expressions—arguments beyond the reach of many middle-grade students.  This process can seem very remote and mysterious to many students. Lacking an understanding of the underlying rationale, many students are therefore unable to repair their errors and clear up their confusions about division of fractions on their own. An alternate approach involves helping students . . . understand the meaning of division as repeated subtraction.”  -PSSM, page 219   (underline added)

For the sophisticated arguments, see pages 2-3 in Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education, By H.Wu

PSSM recommends that probability be covered in every grade, offering five PSSM sections on “Data Analysis and Probability”.  The NCTM is not bothered by the fact that any meaningful discussion of elementary probability requires prior mastery of fractions.

Liping Ma’s Book: U.S. Elementary School Teachers Don’t Understand Arithmetic

U.S. elementary school teachers frequently don’t understand the underlying “whys” of arithmetic, but the same can’t be said of Chinese math teachers.  This is one message of the new book,  Knowing and Teaching Elementary Mathematics (KTEM) by Liping Ma.  (Please see Roger Howe’s Review of Liping Ma’s Book  and  Richard Askey’s Review of Liping Ma’s Book).

U.S. teachers fared poorly when asked questions related to the teaching of:

  1. Subtraction with regrouping
  2. Multidigit multiplication
  3. Dividing fractions

Don’t blame the teachers!  None of these topics appear in PSSM or the PSSM Draft.  The term “subtraction with regrouping (or renaming)” is never used.  There is one example of multidigit multiplication, found on page 220 of PSSM. The text states: “the cost of 1.37 pounds of cheese at $2.95 a pound might be estimated, although a calculator would probably be the preferred tool.”  The failure to cover division by fractions was discussed above.

Liping Ma’s book is referenced in two ways in PSSM.  In each case her ideas have been misused:

  1. Ma uses the phrase “profound understanding of fundamental mathematics” (PUFM).  A teacher who possesses PUFM has a comprehensive understanding of the “network of procedural and conceptual topics” that comprise elementary mathematics. Such a teacher “is able to reveal and represent connections among mathematical concepts and procedures to students.” -Ma, Page 124
    • Ma is referenced on page 17 of PSSM, where we are told that teachers who know “fractions can be understood as parts of a whole, the quotient of two integers, or a number on a line” have an understanding that may be characterized as ‘profound understanding of fundamental mathematics’ (Ma, 1999).”
      • This is not an illustration of PUFM. It is an example of a basic learning expectation for all students.
  2. The terms “compose” and “decompose” appear frequently in PSSM and in KTEM (but not in the PSSM Draft, which preceded the publication of KTEM). In KTEM these words are only used relative to place value.  If “compose” appears alone in KTEM, it’s always shorthand for “compose a unit of higher value” (old term is carrying).  If “decompose” appears alone in KTEM, it’s always shorthand for “decompose a unit of higher value” (old term is borrowing).  PSSM never uses these terms this way. PSSM never discusses carrying or borrowing  (or any other equivalent terms).

The NCTM says in boldface  “The status quo of traditional mathematics isn’t working.”  (See the second fact sheet at NCTM Speaks Out: Setting the Record Straight about changes in Mathematics Education.)   Liping Ma’s book shows that the real problem is the failure to correctly teach “traditional mathematics.”

Not Subtle in Connecticut:  Arithmetic is Obsolete

It’s difficult to see what’s missing, but in Connecticut it’s in boldface.  The Connecticut Mastery Test (CMT), Third Generation, Mathematics Handbook states ( pages 5 – 6 ):

  • 4th graders will continue not to be expected to demonstrate pencil-and-paper mastery of:
    • subtraction with regrouping.
  • 6th graders will continue not to be expected to demonstrate pencil-and-paper mastery of:
    • addition and subtraction of numbers greater than 10,000 or money amounts greater than $100;
    • multiplication and division by 2-digit or larger factors or divisors;
    • addition and subtraction of fractions with unlike denominators; and
    • computation with non-money decimals.
  • 8th graders will continue not to be expected to to demonstrate pencil-and-paper mastery of:
    • addition and subtraction of numbers greater than 10,000 or money amounts greater than $100;
    • addition and subtraction of fractions, except halves and thirds or when one denominator is a factor of the other; and
    • division with fractions or mixed numbers.

Connecticut teachers are told they have:

Permission to Omit. An amazing amount of time and energy is still expended by you and by your students on increasingly obsolete skills.  Teachers need to give each other permission to skip textbook pages that no longer serve a useful purpose.  So give yourself and your colleagues permission to omit such things as:

      • pencil and paper multiplication problems with two-digit or larger factors (3 digits by 1 digit should be enough);
      • paper and pencil division problems with two-digit or larger divisors (4 digits by 1 digit should be enough); and
      • computation with fractions with unreasonable denominators like sevenths or 11ths (halves, fourths, eighths; thirds and sixths; fifths and tenths should be enough).


Update March 2002: A toned-down version of “Permission to Omit” is now found at the top of page 7 via this link.

Steven Leinwand, mathematics consultant for the Connecticut Department of Education,  wrote  “I believe that CT’s expectations are in fact aligned with the NCTM Standards – both old and new.  However, since these Standards cover grade bands and tend to be more general that our test specifications, it is often difficult to do a direct correlation.”
-email message to wgquirk@wgquirk.com, August 2, 2000

Don’t expect any change in the 13 NSF-sponsored “Standards based” math programs.  Their promoters will reason similarly.

Role of Mathematicians: Advice solicited,  Advice Received, Advice Ignored

The NCTM solicited advice from mathematicians:

In order to provide for this complex advisory function, the NCTM petitioned each of the professional organizations of the Conference Board of the Mathematical Sciences (CBMS) to form an Association Review Group (ARG) that would respond, in stages, to a series of substantial and focused questions framed by the Principles and Standards writing group in the course of its work.   -PSSM, Page xv

The NCTM received excellent input (see examples below), but ignored it. None of the several response reports (including the two quoted below) are referenced in either PSSM or the PSSM Draft.

One set of questions, via a letter from Joan Ferrini-Mundy and Mary Lindquist on April 1, 1997, asked:

  1. What is meant by “algorithmic thinking”?
  2. How should the Standards address the nature of algorithms in their more general mathematical context?
  3. How should the Standards address the matter of invented and standard algorithms for arithmetic computation?
  4. What is it about the nature of algorithms that might be important for children to learn?

Roger Howe, Professor of Mathematics at Yale University, responded in the American Mathematical Society NCTM2000 Association Resource Group Second Report in June, 1997 (See following first report via pdf file at Reports of AMS ARG ). Here are five excerpts from Professor Howe’s response:

An important feature of algorithms is that they are automatic and do not require thought once mastered. Thus learning algorithms frees up the brain to struggle with higher level tasks.  On the other hand, algorithms frequently embody significant ideas, and understanding of these ideas is a source of mathematical power.  -Howe, Page 273

 . . . we suspect it is impractical to ask all children personally to devise an accurate, efficient, and general method for dealing with addition of any numbers—even more so with the other operations. Therefore, we hope that experimental periods during which private algorithms may be developed would be brought to closure with the presentation of and practice with standard algorithms. Also we hope care would be taken to ensure that time spent developing and testing private algorithms will not significantly slow overall progress. -Howe, Page 274

Standard algorithms may be viewed analogously to spelling:  to some degree they constitute a convention, and it is not essential that students operate with them from day one or even in their private thinking; but eventually, as a matter of mutual communication and understanding, it is highly desirable that everyone (that is nearly everyone—we recognize that there are always exceptional cases) learn a standard way of doing the four basic arithmetic operations. -Howe, Page 275

We do not think it wise for students to be left with untested private algorithms for arithmetic operations—such algorithms may only be valid for some subclass of problems.  The virtue of standard algorithms—that they are guaranteed to work for allproblems of the type they deal with—deserves emphasis.  -Howe, Page 275   (bold added)

We would like to emphasize that the standard algorithms of arithmetic are more than just “ways to get the right answer”—that is, they have theoretical as well as practical significance.  For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not be accident, but by virtue of construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials. The division algorithms is also significant for later understanding of real numbers. -Howe, Page 275

Kenneth Ross, Professor of Mathematics, University of Oregon, also responded to these four questions in The Mathematical Association of America’s Second Report from the Task Force.  Here are three excerpts from Professor Ross’s June 17, 1997 report:

The NCTM Standards emphasize that children should be encouraged to create their own algorithms, since more learning results from “doing” rather than “listening” and children will “own” the material if they create it themselves. We feel that this point of view has been over-emphasized in reaction to “mindless drills.”  It should be pointed out that in other activities in which many children are willing to work hard and excel, such as sports and music, they do not need to create their own sports rules or write their own music in order to “own” the material or to learn it well. In all these areas, it is essential for there to be a common language and understanding. Standard mathematical definitions and algorithms serve as a vehicle of human communication. In constructivistic terms, individuals may well understand and visualize the concepts in their own private ways, but we all still have to learn to communicate our thoughts in a commonly acceptable language. -Ross, Page 1

The starting point for the development of children’s creativity and skills should be established concepts and algorithms. As part of the natural encouragement of exploration and curiosity, children should certainly be allowed to investigate alternative approaches to the task of an algorithm. However, such investigation should be viewed as motivating, enriching, and supplementing standard approaches. Success in mathematics needs to be grounded in well-learned algorithms as well as understanding of the concepts. None of us advocates “mindless drills.”  But drills of important algorithms that enable students to master a topic, while at the same time learning the mathematical reasoning behind them, can be used to great advantage by a knowledgeable teacher. Creative exercises that probe students’ understanding are difficult to develop but are essential.  -Ross, Page 1-2

The challenge, as always, is balance. “Mindless algorithms” are powerful tools that allow us to operate at a higher level. The genius of algebra and calculus is that they allow us to perform complex calculations in a mechanical way without having to do much thinking. One of the most important roles of a mathematics teacher is to help students develop the flexibility to move back and forth between the abstract and the mechanical. Students need to realize that, even though part of what they are doing is mechanical, much of mathematics is challenging and requires reasoning and thought. -Ross, Page 2

Published in American Educator (AFT), But Also Ignored by the NCTM

Below you will find links to two articles that were published in the Fall 1999 issue of American Educator/American Federation of Teachers.  Both of these were available well prior to the release of PSSM.  Neither is referenced in PSSM.

Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education, By H.Wu, Professor of Mathematics, University of California, Berkeley

Please see Professor Wu’s discussion of the division of fractions and the standard algorithms.  The following two quotes are from the beginning of Professor Wu’s article.

Education seems to be plagued by false dichotomies. Until recently, when research and common sense gained the upper hand, the debate over how to teach beginning reading was characterized by many as “phonics vs. meaning.”  It turns out that, rather than a dichotomy, there is an inseparable connection between decoding—what one might call the skills part of reading—and comprehension. Fluent decoding, which for most children is best ensured by the direct and systematic teaching of phonics and lots of practice reading, is an indispensable condition of comprehension. -Wu, Page 1

“Facts vs. higher order thinking” is another example of a false choice that we often encounter these days, as if thinking of any sort—high or low—could exist outside of content knowledge. In mathematics education, this debate takes the form of “basic skills or conceptual understanding.” This bogus dichotomy would seem to arise from a common misconception of mathematics held by a segment of the public and the education community: that the demand for precision and fluency in the execution of basic skills in school mathematics runs counter to the acquisition of conceptual understanding. The truth is that in mathematics, skills and understanding are completely intertwined. In most cases, the precision and fluency in the execution of the skills are the requisite vehicles to convey the conceptual understanding. There is not ‘conceptual understanding’ and ‘problem-solving skill’ on the one hand and ‘basic skills’ on the other. Nor can one acquire the former without the latter. – Wu, Page 1

 Knowing And Teaching Elementary Mathematics, By Richard Askey, John Boscom Professor of Mathematics, University of Wisconsin-Madison

The title of this article is also the title of the new book by Liping Ma. Please see Professor Askey’s discussion of the division of fractions.  The following quote is from Professor Askey’s article.

As the word ‘understanding’ continues to be bandied about loosely in the debates over math education, this book provides a much-needed grounding. It disabuses people of the notion that elementary school mathematics is simple—or easy to teach. It cautions us, as Ma says in her conclusion, that  ‘the key to reform…[is to] focus on substantive mathematics.‘  And at the book’s heart is the idea that student understanding is heavily dependent on teacher understanding.  -Askey, Page 2
(Bold emphasis added)

Next?


Copyright 2000-2002 William G. Quirk, Ph.D.

The Road to Building Critical Mass, Or How to Bring Real Change to U.S. Mathematics Education

The Road to Building Critical Mass, Or How to Bring Real Change to U.S. Mathematics Education

Nakonia (Niki) Hayes Columnist EdNews.org

First, Politics Require Understanding and Use
Mathematics education reformists warn us to keep politics out of the math “debates.” That’s a lofty and ideal goal. Political actions too often cause more problems than they solve as people see their issues as “wins” and “losses” that support egos and turf.

But keeping politics out of education issues, including mathematics, is no longer a reality. Putting the right people to work in that world is the tricky part. A successful program in any profession needs qualified individuals “working the program.” The arrogance has been, for years, that only the reformists had “qualified” people to talk about mathematics education. Imagine their surprise to learn there are those on the other side of the argument who are equally qualified to speak on the issues.

In the meantime, reformists have used their powerful political allies, originally at the National Science Foundation, to help fund the disastrous modern history of mathematics education. It is reasonable to conclude that politics must be embraced by those who oppose the NSF programs and who want real, successful curricula of numeracy.

Second, Look for a Model

A model is offered that is now confronting the disaster of whole language and its effects on reading literacy. It is only because of new, major political muscle that reading literacy is being rescued from the reformists’ methodologies that produced a generation (or more) of non-spellers and illiterate readers. This literacy recovery has been achieved because Pres. George Bush was joined by Sen. Ted Kennedy in sponsoring the No Child Left Behind legislation. That legislation included billion-dollar funding to support a return to proven scientific teaching methods of reading, and develop critical masses of educators and students.

Called Reading First, the program can give insight on how power players in the mathematics conflicts can achieve success. There power players exist in mathematics, mathematics education, businesses, and among legislators who want a balance of math instruction that includes both conceptual understanding (the reformists’ program) and principles/basic skills (the traditionalists’ program).

Third, Learn from the Model—Save Time, Energy, and Dollars
The steps in this model are explained by Sol Stern in the City Journal, Winter 2007, in an article entitled “This Bush Education Reform Really Works.” Its subtitle is “Reading First, much maligned, succeeds in teaching kids to read.” The summarized points have been written to reflect “mathematics education”:

1. Secure powerful federal support with a specific goal of solving, not just addressing, the problems in mathematics education. This will be needed to counter the billions spent by the National Science Foundation and private donors who fund reformist math programs.

* Reading First has had a budget of $1 billion per year as part of the NCLB legislation. Reid Lyon, chief reading scientist for The National Institute of Child Health and Human Development (NICHD), and Robert Sweet of the House education committee, drafted the Reading First legislation early in 2001. They consciously designed Reading First to do an end run around the deeply entrenched whole-language movement.

Sweet said, “Reading First was created to be a catalyst, to provide a financial incentive for schools finally to start doing the right thing for the millions of kids left behind in reading.”

* Said writer Stern, “You could say that Reading First was a $6 billion
federal bribe to get districts to do what they really should have been doing already.”

2. Watch the National Math Advisory Panel: They have reportedly “set themselves a huge and appropriate task of rigorously evaluating all the research available, identifying facts, opinions, and old wives tales,” according to an e-mail today from Dave Myerson in Mercer Island, WA. This action is highlighted in Stern’s article:

* Gather published, peer-reviewed studies that describe not just how
children learn mathematics but why so many fall behind—and how schools can best keep it from happening.

* Studies by the National Institute of Child Health and Human Development (NICHD), a wing of the National Institutes of Health, sponsored reading research at universities, with scientists from cognitive neuroscience, pediatrics, genetics, educational psychology, and child development.

3. Face an instituted money hurdle and name it: Science will collide head-on with ideologies and economic interests within the halls of public education. Interests other than pedagogical are at stake. A shift in teaching methodologies will put tenured jobs and professional development contracts from the $500 billion-plus education industry up for grabs.

4. Face another money hurdle and name it: Progressive classroom instruction is promulgated by the education schools that monopolize teacher training. Education schools do not produce money for universities, so grants to promote ideologies bring money into the coffers.

5. Learn euphemisms: Morphed descriptions of progressive education terminology are designed to make programs sound more reasonable to dubious parents.

6. Design shields against open hostility to science in the education industry: Nonetheless, demand scientific research that supports reform math programs. Fight attempts from reformists who demand “implementation of diverse kinds of scientific research, including teacher research. (Teachers evaluate instructional methods by observing their own classrooms; science be damned.)

7. Use respected sources for surveys: Secure assessment through the National Council on Teacher Quality the percent of elementary education classes that don’t teach the principles of mathematics and scientific math instruction. (For example, 85% of surveyed ed schools showed elementary ed classes don’t teach principles of phonics and scientific reading instruction.)

8. Use winning strategies, with no shame: Consciously plan an end-run around the deeply entrenched whole-math ideology, even with limited power sources.

9. Be prepared to compromise, but on your terms: Reading First legislation abandoned the idea of requiring participating districts to use only scientifically tested reading programs. Instead, districts could also use untested ones, as long as they adhered to the principles of scientific reading instruction: phonemic awareness, phonics, vocabulary, fluency, and comprehension.

* In a book review by Bill Evers of Class Warfare: Besieged Schools,
Bewildered parents, Betrayed Kids and the Attack on Excellence, written by J. Martin Rochester, this idea of compromise (“a potential middle ground”) is addressed:

* “…traditional education (solid content, drill and practice, teacher-led
classrooms) modified by some of the defensible ideas of progressive education (emphasis on motivation, critical thinking, some projects, some field trips).”

But, Rochester “is realistic in saying that is basis-plus compromise may be difficult to achieve in practice,” writes Evers. That means the idea of basic-plus compromise must be held firmly. (Does that make it not a compromise?)

* Special note on Bill Evers: He has just been nominated by Pres. Bush to be the U.S. Assistant Secretary of Education.

10. Use new dollars to push for a critical mass: First critical mass will be composed of schools who sign on to the program, in order to ignite a countercultural education movement of teachers, parents, administrators, and education activists.

* By 2006, there were 5,600 schools in 1,700 school districts nationwide who have received Reading First grants.

11. Push for a different critical mass with teacher training. 
*  By 2006, there have been 100,000 K-3 teachers who have received/are receiving professional development in reading science. This has removed these early childhood teachers from the ed schools’ ideological orbit.

12. Set a comprehensive study by an outside evaluator at the end of 5 years: Record those who have succeeded and those who haven’t, including those who have used the program and those who have not. Be clear about those who follow the progressive, ideological methods, and those who enlist for the money but do not show good faith or fidelity to the program.

13. Reprioritize funding due to congressional oversight: More financial help needs to go to places that have really embraced scientific math instruction, are getting strong results, and are truly needy.

14. Name those who refuse opportunities: Openly name education leaders in states and cities who, offered the solution, didn’t grab it.

Published February 15, 2007

 

Testimony of David Klein

Testimony of David Klein
Professor of Mathematics
California State University, Northridge

 

April 4, 2000

 

U.S. House of Representatives Committee on Appropriations
Subcommittee on Labor, Health and Human Services, Education and Related Activities

 


Thank you for the opportunity to testify before you today.

Last October, the U.S. Department of Education released a report to the nation’s 16,000 school districts. The report designated 10 mathematics programs for K-12 as “exemplary” or “promising.” The following month, I sent an open letter co-authored with mathematicians Richard Askey, R. James Milgram, and Hung-Hsi Wu with more than 200 other co-signers to Education Secretary Richard Riley urging him to withdraw the Department of Education’s recommendations (http://www.mathematicallycorrect.com/riley.htm).

Among the endorsers are many of the nation’s most accomplished scientists and mathematicians. Department heads at many universities, including Caltech, Stanford, Harvard, and Yale, along with two former presidents of the Mathematical Association of America also added their names in support. Seven Nobel laureates and winners of the Fields Medal, the highest award in mathematics, also endorsed. In addition, several prominent state and national education leaders co-signed our open letter.

The ten so-called “exemplary” and “promising” math programs recommended by the Department of Education for our children include some of the worst math books available. The programs I have examined radically de-emphasize basic skills in arithmetic and algebra. Uncontrolled calculator use is rampant. One can draw a parallel between the philosophy that underlies the failed “whole language learning” approach to reading, and the Department of Education’s agenda for mathematics.

The effects of this philosophy in Los Angeles have been devastating. According to a recent Los Angeles Times article(3/17/2000): “Sixty percent of the eighth-graders in L.A. Unified, it is estimated, do not yet know their multiplication tables.”

Proponents of these watered-down programs believe that they are appropriate for minority students and women. For example, Jack Price, one of the Expert Panel members for the Department of Education, said on a radio show in 1996 (http://mathematicallycorrect.com/roger.htm) that minority groups and women do not learn math the same way as white males. He stated:

 

… women have a tendency to learn better in a collaborative effort when they are doing inductive reasoning.

This was in contrast to the way white males learn math. According to Jack Price, “males … learn better deductively in a competitive environment.

This misguided view of women and minorities is consistent with the Department of Education’s math books. They rely heavily on superficial repetitive patterns to draw conclusions rather than logical deduction, which is the core of mathematics.

It is true that the National Council of Teachers of Mathemtics, or NCTM, has endorsed the Department of Education’s list of math books, and the National Science Foundation has spent millions of dollars to develop and promote several of them.

But these organizations represent surprisingly narrow interests, and there is a revolving door between them.

Steven Leinwand, a co-chair of the Expert Panel, was also a member of the Advisory Boards for two programs found to be “exemplary” by his panel. He also serves on the Board of Directors of the NCTM. Expert Panel member, Jack Price, who I just quoted, is a former president of the NCTM.

Luther Williams, who as Assistant Director of the NSF approved the funding of several of the recommended curricula, also served on the Expert Panel. Glenda Lappan, the current president of the NCTM, is a co-author of the so-called “exemplary” program, Connected Mathematics, which her organization endorses.

These facts suggest obvious conflicts of interest. According to the official minutes of the Department of Education’s Expert Panel, the panel members were themselves aware of this conflict.

I quote from the November 15 and 16, 1996 minutes of the second meeting of the Expert Panel.

 

Some members expressed their concern about serving as chair, because their organizations develop products that may be reviewed by the panel, and they were concerned about conflict of interest. The panel agreed that if conflict of interest were an issue, it would not matter whether one served as chair or simply as a member. The panel agreed that whoever is panel chair ought to be able to act independently of his or her own interests.

I urge the distinguished members of the Appropriations Committee to investigate the possibility of conflict of interest within the Expert Panel.

I also urge the distinguished members of this committee not to allocate funds to promote mathematics programs premised on the misguided notion that women and minority groups need a different and inferior kind of mathematics.

 


TERC Hands-On Math: A Snapshot View

TERC Hands-On Math:  A Snapshot View

 by Bill Quirk  ( wgquirk@wgquirk.com)

Developed by TERC, with funding from the National Science Foundation (NSF),  Investigations in Number, Data, and Space  purports to be “a complete K-5 mathematics curriculum that supports all students as they learn to think mathematically.” The NSF is now spending millions to promote implementation of the TERC program.  School Boards find it difficult to say no. They rationalize: “it’s just a different way to teach elementary math, and the NSF backs it, so how bad can it be?”

This program is very bad because it omits standard computational methods, standard formulas, and standard terminology.  TERC says this is now obsolete, due to the power of $5 calculators.  They claim their program moves “beyond arithmetic” to offer “significant math,” including important ideas from probability, statistics, 3-D geometry, and number theory.

But math is a vertically-structured knowledge domain.  Learning more advanced math isn’t possible without first mastering traditional pencil-and-paper arithmetic. This truth is clearly demonstrated by the shallow details of the TERC fifth grade program.  Their most advanced “Investigations” offer probability without multiplying fractions, statistics without the arithmetic mean, 3-D geometry without formulas for volume, and number theory without prime numbers.

TERC Omits All Standard Computational Methods

Consider the “Sample of Ads Investigation,” at the end of the TERC fifth grade.  Students are given a 48-page newspaper and a supply of  “Recording Strips” that are premarked with “familiar fractions,” such as 1/4 and 2/3.  They begin by deciding to sample one-third of the 48 pages.  After using a calculator to divide 48 by 3, they select 16 sample pages and use eyeball estimation to guess the fraction of ads found on each sample page.  Then, using one 3-inch “Recording Strip” for each sample page, students color the fraction of ads, cut out the colored portions, and tape them onto a 48-inch length of adding machine tape, “starting from one end of the tape and putting the pieces right next to each other.”  Students then estimate the fraction of ads for the full 16-page sample by folding the 48-inch strip to estimate the fraction corresponding to the 16 colored-in pieces.

Why not add the 16 fractions and then divide the sum by 16?  TERC students never learn about dividing fractions, and they never learn general methods for adding fractions.  They do learn a hands-on method for adding two proper fractions with denominators less than 7, but this paper-folding method doesn’t work if the denominator of the sum fraction isn’t also less than 7.

TERC Omits Standard Formulas

For the final fifth grade Investigation in 3-D geometry, TERC students use patterns, scissors, and tape to build prisms, pyramids, cylinders, and cones.  They then attempt to “discover” 3 to 1 volume relationships by pouring rice from one bulging container into another.  Later they find the volume of each paper container by pouring rice from the container into a  plastic measuring tool.

Why not formulas for volume?  TERC says students usually don’t understand formulas and frequently apply them blindly and incorrectly.  So general methods involving standard formulas are not found in TERC math.

TERC Omits Standard Terminology

TERC recommends natural language, not standard terminology.    For statistics they say “we have found useful such words as clumps, clusters, bumps, gaps, holes, spread out, and bunched together.”   For “the  mathematics of change” they recommend “grow, shrink, faster, slower, steep, flat, slow, steady, speed up, slow down, grows steadily, grows faster and faster, grows slower and slower, shrinks steadily, shrinks slower and slower, shrinks faster and faster, grows and then shrinks, oscillates between growing and shrinking.”  TERC appears to believe that these subjective terms are related to calculus.

TERC has redefined the meaning of “think mathematically” and painted a false picture of elementary mathematics.  It’s all very hard to believe, but it’s sadly true.

For a detailed analysis of TERC math, click on TERC Hands-On Math: The Truth is in the Details.

 


Next?


Copyright 2002-2005 William G. Quirk, Ph.D.

 

Teacher can’t teach

Teacher can’t teach

Over the past half-century, the number of pupils in U.S. schools grew by about 50 percent while the number of teachers nearly tripled. Spending per student rose threefold, too. If the teaching force had simply kept pace with enrollments, school budgets had risen as they did, and nothing else changed, today’s average teacher would earn nearly $100,000, plus generous benefits. We’d have a radically different view of the job and it would attract different sorts of people.

Yes, classes would be larger—about what they were when I was in school. True, there’d be fewer specialists and supervisors. And we wouldn’t have as many instructors for youngsters with “special needs.” But teachers would earn twice what they do today (less than $50,000, on average) and talented college graduates would vie for the relatively few openings in those ranks.

What America has done, these past 50 years, is invest in more teachers rather than better ones, even as countless appealing and lucrative options have opened up for the able women who once poured into public schooling. No wonder teaching salaries have just kept pace with inflation, despite huge increases in education budgets. No wonder the teaching occupation, with blessed exceptions, draws people from the lower ranks of our lesser universities. No wonder there are shortages in key branches of this sprawling profession. When you employ three million people and you don’t pay very well, it’s hard to keep a field fully staffed, especially in locales (rural communities, tough urban schools) that aren’t too enticing and in subjects such as math and science where well-qualified individuals can earn big bucks doing something else.

Why did we triple the size of the teaching work force instead of paying more to a smaller number of stronger people? Three reasons.

First, the seductiveness of smaller classes. Teachers want fewer kids in their classrooms and parents think their children will be better off, despite scant evidence that students learn more in smaller classes, particularly from less able instructors. Second, the institutional interests that benefit from a larger teaching force, above all dues-collecting (and influence-seeking) unions, and colleges of education whose revenues (tuition, state subsidies) and size (all those faculty slots) depend on their enrollments. Third, the social forces pushing schools to treat children differently from one another, creating one set of classes for the gifted, others for children with handicaps, those who want to learn Japanese, who seek full-day kindergarten or who crave more community-service opportunities.

Nobody has resisted. It was not in anyone’s interest to keep the teaching ranks sparse, while many interests were served by helping them to swell. Today, we pay the price: lots of money spent on schooling, nearly all of it for salaries, but schooling that, at the end of the day, depends on the knowledge, skills and commitment of teachers who don’t earn much and cannot see that they ever will.

Compounding that problem, we make multiple policy blunders. We restrict entry to people “certified” by state bureaucracies, normally after passing through quasi-monopolistic training programs that add little value. Thus an ill-paid vocation also has daunting, yet pointless, barriers to entry. We pay mediocre instructors the same as super-teachers. Though tiny cracks are appearing in the “uniform salary schedule,” in general an energized and highly effective classroom practitioner earns no more than a feckless time-server. We pay no more to high-school physics or math teachers than middle-school gym teachers, though the latter are easy to find while people capable of the former posts are scarce and have plentiful options. We pay no more to those who take on daunting assignments in tough schools than to those who work with easy kids in leafy suburbs. In fact, we often pay them less.

Instead of recognizing that today’s 20-somethings commonly try multiple occupations before settling down (if they ever do), then making imaginative use of those who are game to teach for a few years, we still assume that teaching is a lifelong vocation and lament anyone who exits the classroom for other pursuits. Instead of deploying technology so that gifted teachers reach hundreds of kids while others function more like tutors or aides, we assume that every classroom needs its own Socrates.

Despite all that, and to their great credit, most teachers are decent folks who care about kids and want to help them learn. But turning around U.S. schools and “leaving no child behind” calls for more. It also requires passion, brains, knowledge and technique. Federal law now demands subject-matter mastery. Such qualities are hard to find in vast numbers, however, especially when the job doesn’t pay very well. Yet fat across-the-board raises for three million people are a pipe dream. (Adding $10,000 plus benefits to their pay would add some $40 billion to school budgets.)

Maybe we can’t turn back the clock on the numbers, but surely we can reverse the policy errors. With hundreds of thousands of teaching jobs now turning over each year, at minimum we should insist that new entrants play by different rules that reward effectiveness, deploy smart incentives and suitable technology, compensate them sensibly, and make skillful use of short-termers instead of just wishing they’d stay longer. And this time let’s watch what we’re doing.

This article originally appeared in the March 11, 2005 edition of the Wall Street Journal. The March 22, 2005 edition published several letters in response, available here (subscription required).

by Chester E. Finn, Jr

Statewide Testing a Step We need to know how students compare


Statewide Testing a Step
We need to know how students compare

– Robert L. Trigg
Wednesday, July 7, 1999

PARENTS, TAXPAYERS, teachers, employers and the nearly 6 million students who attend California’s public schools deserve to know how effective those schools are — and how they compare to schools around the country.

It’s not enough to judge a school by its dropout rate, the percentage of parents who check “very satisfied” on surveys, or the number of computers in each classroom.

We need to know how well students can read, write and do math. We need to know how well students are mastering history, civics and science. We have a right to know whether the billions of dollars we invest in education each year are well spent.

Above all, we owe our children an honest accounting of the schools charged with preparing them for a competitive world.

The state’s assessment program and reform efforts, including a standardized test called the Stanford 9 (SAT 9), help provide the accounting. Public schools are required to give the SAT 9 to students in grades two through 11.

While no test is perfect, the SAT 9 is a solid, reliable means of measuring student achievement against a national norm. The test results are reported individually, as well as by school and district, enabling administrators, teachers, parents and students to understand how well schools and students are doing in relation to students around the state and nation.

The new test is only one part of a broad effort to improve California’s schools. Other efforts include: — Adopting rigorous academic standards in language arts, math, history-social science and science, in keeping with a mandate from the state Legislature. For the first time in memory, the State Board of Educations has spelled out what every student should learn in these areas at each grade level. In most cases, the new standards exceed what districts previously expected students to learn, so schools around the state are going through the difficult process of raising the bar for all students. — Bringing together with the leadership of Governor Gray Davis broad political support for the Public School Accountability Act, which will bring real rewards and consequences to schools based on student achievement. — Requiring, beginning in 2004, that California students will have to pass a high school exit exam in order to graduate. — Lowering class size, approving new and better textbooks, boosting beginning teachers’ salaries, and funding additional teacher training in order to support all of this change.

California’s new academic standards reflect what students will need to know in a 21st century world. While few students enjoy taking standardized tests, the SAT 9 will help keep school officials and teachers focused on our tough standards. If a school is going to be judged, in part, on its test scores, teachers must focus on what is on the test. The SAT 9 has been criticized because the material it covers does not precisely match the material the state’s teachers are now required to teach. The reality is that no nationally normed test is perfectly aligned with our new standards. This year, in accordance with the legislation establishing the new testing program, we added “augmentation items” to the test in the areas of language arts and math. These items will not provide national comparisons, but they will directly test student mastery of our new standards. These augmentation items have been criticized because they test material that, in some cases, hasn’t yet been incorporated into schools’ curricula. While I sympathize with teachers and students wanting to do their best, I believe it’s important to move ahead with this part of the testing program. This year’s results will give us baseline data. California has waited years for a statewide testing program. Without question, we should continue to refine and improve the SAT 9. The State Board of Education welcomes constructive criticism. We will not abandon statewide testing simply because the test isn’t perfect. No standardized test will ever be perfect.

Standardized tests are one of the best defenses our children have against inferior education. Let’s stay the course and build the best public school system California has ever had.

Robert L. Trigg is a former school superintendent and has served on the State Board of Education since 1996. UNDERSET 3 INs

Page A – 19
URL: http://sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/1999/07/07/ED58584.DTL


©2005 San Francisco Chronicle

The untold story behind the famous rise — and shameful fall — of Jaime Escalante, America’s master math teacher.

Stand and Deliver Revisited       July 2002

The untold story behind the famous rise — and shameful fall — of Jaime Escalante, America’s master math teacher.


By Jerry Jesness

 

Thanks to the popular 1988 movie Stand and Deliver, many Americans know of the success that Jaime Escalante and his students enjoyed at Garfield High School in East Los Angeles. During the 1980s, that exceptional teacher at a poor public school built a calculus program rivaled by only a handful of exclusive academies.

It is less well-known that Escalante left Garfield after problems with colleagues and administrators, and that his calculus program withered in his absence. That untold story highlights much that is wrong with public schooling in the United States and offers some valuable insights into the workings — and failings — of our education system.

Escalante’s students surprised the nation in 1982, when 18 of them passed the Advanced Placement calculus exam. The Educational Testing Service found the scores suspect and asked 14 of the passing students to take the test again. Twelve agreed to do so (the other two decided they didn’t need the credit for college), and all 12 did well enough to have their scores reinstated.

In the ensuing years, Escalante’s calculus program grew phenomenally. In 1983 both enrollment in his class and the number of students passing the A.P. calculus test more than doubled, with 33 taking the exam and 30 passing it. In 1987, 73 passed the test, and another 12 passed a more advanced version (“BC”) usually given after the second year of calculus.

By 1990, Escalante’s math enrichment program involved over 400 students in classes ranging from beginning algebra to advanced calculus. Escalante and his fellow teachers referred to their program as “the dynasty,” boasting that it would someday involve more than 1,000 students.

That goal was never met. In 1991 Escalante decided to leave Garfield. All his fellow math enrichment teachers soon left as well. By 1996, the dynasty was not even a minor fiefdom. Only seven students passed the regular (“AB”) test that year, with four passing the BC exam — 11 students total, down from a high of 85.

In any field but education, the combination of such a dramatic rise and such a precipitous fall would have invited analysis. If a team begins losing after a coach is replaced, sports fans are outraged. The decline of Garfield’s math program, however, went largely unnoticed.

Movie Magic

Most of us, educators included, learned what we know of Escalante’s experience from Stand and Deliver. For more than a decade it has been a staple in high school classes, college education classes, and faculty workshops. Unfortunately, too many students and teachers learned the wrong lesson from the movie.

Escalante tells me the film was 90 percent truth and 10 percent drama — but what a difference 10 percent can make. Stand and Deliver shows a group of poorly prepared, undisciplined young people who were initially struggling with fractions yet managed to move from basic math to calculus in just a year. The reality was far different. It took 10 years to bring Escalante’s program to peak success. He didn’t even teach his first calculus course until he had been at Garfield for several years. His basic math students from his early years were not the same students who later passed the A.P. calculus test.

Escalante says he was so discouraged by his students’ poor preparation that after only two hours in class he called his former employer, the Burroughs Corporation, and asked for his old job back. He decided not to return to the computer factory after he found a dozen basic math students who were willing to take algebra and was able to make arrangements with the principal and counselors to accommodate them.

Escalante’s situation improved as time went by, but it was not until his fifth year at Garfield that he tried to teach calculus. Although he felt his students were not adequately prepared, he decided to teach the class anyway in the hope that the existence of an A.P. calculus course would create the leverage necessary to improve lower-level math classes.

His plan worked. He and a handpicked teacher, Ben Jimenez, taught the feeder courses. In 1979 he had only five calculus students, two of whom passed the A.P. test. (Escalante had to do some bureaucratic sleight of hand to be allowed to teach such a tiny class.) The second year, he had nine calculus students, seven of whom passed the test. A year later, 15 students took the class, and all but one passed. The year after that, 1982, was the year of the events depicted in Stand and Deliver.

The Stand and Deliver message, that the touch of a master could bring unmotivated students from arithmetic to calculus in a single year, was preached in schools throughout the nation. While the film did a great service to education by showing what students from disadvantaged backgrounds can achieve in demanding classes, the Hollywood fiction had at least one negative side effect. By showing students moving from fractions to calculus in a single year, it gave the false impression that students can neglect their studies for several years and then be redeemed by a few months of hard work.

This Hollywood message had a pernicious effect on teacher training. The lessons of Escalante’s patience and hard work in building his program, especially his attention to the classes that fed into calculus, were largely ignored in the faculty workshops and college education classes that routinely showed Stand and Deliver to their students. To the pedagogues, how Escalante succeeded mattered less than the mere fact that he succeeded. They were happy to cheer Escalante the icon; they were less interested in learning from Escalante the teacher. They were like physicians getting excited about a colleague who can cure cancer without wanting to know how to replicate the cure.

 

The Secrets to His Success

How did Escalante attain such success at Garfield? One key factor was the support of his principal, Henry Gradillas.

Escalante’s program was already in place when Gradillas came to Garfield, but the new principal’s support allowed it to run smoothly. In the early years, Escalante had met with some resistance from the school administration. One assistant principal threatened to have him dismissed, on the grounds that he was coming in too early (a janitor had complained), keeping students too late, and raising funds without permission. Gradillas, on the other hand, handed Escalante the keys to the school and gave him full control of his program.

Gradillas also worked to create a more serious academic environment at Garfield. He reduced the number of basic math classes and eventually came up with a requirement that those who take basic math must concurrently take algebra. He even braved the wrath of the community by denying extracurricular activities to entering students who failed basic skills tests and to current students who failed to maintain a C average.

In the process of raising academic standards at Garfield, Gradillas made more than a few enemies. He took a sabbatical leave to finish his doctorate in 1987, hoping that upon his return he would either be reinstated as principal of Garfield or be given a position from which he could help other schools foster programs like Escalante’s. He was instead assigned to supervise asbestos removal. It is probably no coincidence that A.P. calculus scores at Garfield peaked in 1987, Gradillas’ last year there.

Escalante remained at Garfield for four years after Gradillas’ departure. Although he does not blame the ensuing administration for his own departure from the school, Escalante observes that Gradillas was an academic principal, while his replacement was more interested in other things, such as football and the marching band.

Gradillas was not the only reason for Escalante’s success, of course. Other factors included:

The Pipeline. Unlike the students in the movie, the real Garfield students required years of solid preparation before they could take calculus. This created a problem for Escalante. Garfield was a three-year high school, and the junior high schools that fed it offered only basic math. Even if the entering sophomores took advanced math every year, there was not enough time in their schedules to take geometry, algebra II, math analysis, trigonometry, and calculus.

So Escalante established a program at East Los Angeles College where students could take these classes in intensive seven-week summer sessions. Escalante and Gradillas were also instrumental in getting the feeder schools to offer algebra in the eighth and ninth grades.

Inside Garfield, Escalante worked to ratchet up standards in the classes that fed into calculus. He taught some of the feeder classes himself, assigning others to handpicked teachers with whom he coordinated and reviewed lesson plans. By the time he left, there were nine Garfield teachers working in his math enrichment program and several teachers from other East L.A. high schools working in the summer program at the college.

Tutoring. Years ago, when asked if Garfield could ever catch up to Beverly Hills High School, Gradillas responded, “No, but we can get close.” The children of wealthy, well-educated parents do enjoy advantages in school. Escalante did whatever he could to bring some of those advantages to his students.

Among the parents of Garfield students, high school graduates were in the minority and college graduates were a rarity. To help make up for the lack of academic support available at home, Escalante established tutoring sessions before and after school. When funds became available, he arranged for paid student tutors to help those who fell behind.

Escalante’s field-leveling efforts worked. By 1987, Gradillas’ prediction proved to be partially wrong: In A.P. calculus, Garfield had outpaced Beverly High.

Open Enrollment. Escalante did not approve of programs for the gifted, academic tracking, or even qualifying examinations. If students wanted to take his classes, he let them.

His open-door policy bore fruit. Students who would never have been selected for honors classes or programs for the gifted chose to enroll in Escalante’s math enrichment classes and succeeded there.

Of course, not all of Escalante’s students earned fives (the highest score) on their A.P. calculus exams, and not all went on to receive scholarships from top universities. One argument that educrats make against programs like Escalante’s is that they are elitist and benefit only a select few.

Conventional pedagogical wisdom holds that the poor, the disadvantaged, and the “culturally different” are a fragile lot, and that the academic rigor usually found only in elite suburban or private schools would frustrate them, crushing their self-esteem. The teachers and administrators that I interviewed did not find this to be true of Garfield students.

Wayne Bishop, a professor of mathematics and computer science at California State University at Los Angeles, notes that Escalante’s top students generally did not attend Cal State. Those who scored fours and fives on the A.P. calculus tests were at schools like MIT, Harvard, Yale, Berkeley, USC, and UCLA. For the most part, Escalante grads who went to Cal State-L.A. were those who scored ones and twos, with an occasional three, or those who worked hard in algebra and geometry in the hope of getting into calculus class but fell short.

Bishop observes that these students usually required no remedial math, and that many of them became top students at the college. The moral is that it is better to lose in the Olympics than to win in Little League, even for those whose parents make less than $20,000 per year.

Death of a Dynasty

Escalante’s open admission policy, a major reason for his success, also paved the way for his departure. Calculus grew so popular at Garfield that classes grew beyond the 35-student limit set by the union contract. Some had more than 50 students. Escalante would have preferred to keep the classes below the limit had he been able to do so without either denying calculus to willing students or using teachers who were not up to his high standards. Neither was possible, and the teachers union complained about Garfield’s class sizes. Rather than compromise, Escalante moved on.

Other problems had been brewing as well. After Stand and Deliver was released, Escalante became an overnight celebrity. Teachers and other interested observers asked to sit in on his classes, and he received visits from political leaders and celebrities, including President George H.W. Bush and actor Arnold Schwarzenegger. This attention aroused feelings of jealousy. In his last few years at Garfield, Escalante even received threats and hate mail. In 1990 he lost the math department chairmanship, the position that had enabled him to direct the pipeline.

A number of people at Garfield still have unkind words for the school’s most famous instructor. One administrator tells me Escalante wanted too much power. Some teachers complained that he was creating two math departments, one for his students and another for everyone else. When Escalante quit his job at Garfield, John Perez, a vice president of the teachers union, said, “Jaime didn’t get along with some of the teachers at his school. He pretty much was a loner.”

In addition, Escalante’s relationship with his new principal, Maria Elena Tostado, was not as good as the one he had enjoyed with Gradillas. Tostado speaks harshly about her former calculus teachers, telling the Los Angeles Times they’re disgruntled former employees. Of their complaints, she said, “Such backbiting only hurts the kids.”

Escalante left the program in the charge of a handpicked successor, fellow Garfield teacher Angelo Villavicencio. Escalante had met Villavicencio six years previously through his students — he had been a math teacher at Griffith Junior High, a Garfield feeder. At Escalante’s request and with Gradillas’ assistance, Villavicencio came to Garfield in 1985. At first he taught the classes that fed into calculus; later, he joined Escalante and Ben Jimenez in teaching calculus itself.

When Escalante and Jimenez left in 1991, Villavicencio ascended to Garfield’s calculus throne. The following year he taught all of Garfield’s AB calculus students — 107 of them, in two sections. Although that year’s passing rate was not as high as it had been in previous years, it was still impressive, particularly considering that two-thirds of the calculus teachers had recently left and that Villavicencio was working with lecture-size classes. Seventy-six of his students went on to take the A.P. exam, and 47 passed.

That year was not easy for Villavicencio. The class-size problem that led to Escalante’s departure had not been resolved. Villavicencio asked the administration to add a third section of calculus so he could get his class sizes below 40, but his request was denied. The principal attempted to remove him from Music Hall 1, the only room in the school that could comfortably ac-commodate 55 students. Villavicencio asked himself, “Am I going to have a heart attack defending the program?” The following spring he followed Escalante out Garfield’s door.

Scattered Legacy

When Cal State’s Wayne Bishop called Garfield to ask about the status of the school’s post-Escalante A.P. calculus program, he was told, “We were doing fine before Mr. Escalante left, and we’re doing fine after.” Soon Garfield discovered how critical Escalante’s presence had been. Within a few years, Garfield experienced a sevenfold drop in the number of A.P. calculus students passing their exams. (That said, A.P. participation at Garfield is still much, much higher than at most similar schools. In May of 2000, 722 Garfield students took Advanced Placement tests, and 44 percent passed.)

Escalante moved north to Sacramento, where he taught math, including one section of calculus, at Hiram Johnson High School. He calls his experience there a partial success. In 1991, the year before he began, only six Johnson students took the A.P. calculus exam, all of whom passed. Three years later, the number passing was up to 18 — a respectable improvement, but no dynasty. It had taken Escalante over a decade to build Garfield’s program. Already in his 60s when he made his move, he did not have a decade to build another powerhouse in new territory.

Meanwhile, Villavicencio moved to Chino, a suburb east of Los Angeles. He had to take a pay cut of more than $7,000, since his new school would pay him for only six of his 13 years in teaching. (Like many districts, the Chino Valley Unified School District had a policy of paying for only a limited number of years of outside experience.) In Chino, Villavicencio again taught A.P. calculus, first in Ayala High School and later in Don Lugo High School.

In 1996 he contacted Garfield’s new principal, Tony Garcia, and offered to come back to help revive the moribund calculus program. He was politely refused, so he stayed at Don Lugo. Villavicencio worked with East Los Angeles College to establish a branch of the Escalante summer school program there. This program, along with more math offerings in the district’s middle schools, allowed Villavicencio to admit even some ninth-graders into his calculus class.

After Villavicencio got his program running smoothly, it was consistently producing A.P. calculus passing scores in the 60 percent to 70 percent range. Buoyed by his success, he requested that his salary be raised to reflect his experience. His request was denied, so he decided to move on to another school. Before he left, Don Lugo High was preparing to offer five sections of AB calculus and one section of BC. In his absence, there were only two sections of AB and no BC.

Meanwhile, after seeing its calculus passing rate drop into the single digits, Garfield is experiencing a partial recovery. In the spring of 2001, 17 Garfield students passed the AB calculus exam, and seven passed the BC. That is better than double the number of students passing a few years ago but less than one-third the number passing during the glory years of Escalante’s dynasty.

And after withering in the absence of its founder, the Escalante program at East Los Angeles College has revived. Program administrator Paul Powers reports that over 1,000 high school students took accelerated math classes through the college in the year 2000.

Although the program now accepts students from beyond the college’s vicinity, the target pupils are still those living in East L.A.

Nationally, there is no denying that the Escalante experience was a factor in the growth of Advanced Placement courses during the last decade and a half. The number of schools that offer A.P. classes has more than doubled since 1983, and the number of A.P. tests taken has increased almost sixfold. This is a far cry from the Zeitgeist of two decades ago, when A.P. was considered appropriate only for students in elite private and wealthy suburban public schools.

Still, there is no inner-city school anywhere in the United States with a calculus program anything like Escalante’s in the ’80s. A very successful program rapidly collapsed, leaving only fragments behind.

This leaves would-be school reformers with a set of uncomfortable questions. Why couldn’t Escalante run his classes in peace? Why were administrators allowed to get in his way? Why was the union imposing its “help” on someone who hadn’t requested it? Could Escalante’s program have been saved if, as Gradillas now muses, Garfield had become a charter school? What is wrong with a system that values working well with others more highly than effectiveness?

Barn Building

Lyndon Johnson said it takes a master carpenter to build a barn, but any jackass can kick one down. In retrospect, it’s fortunate that Escalante’s program survived as long as it did. Had Garfield’s counselors refused to let a handful of basic math students take algebra back in 1974, or had the janitor who objected to Escalante’s early-bird ways been more influential, America’s greatest math teacher might just now be retiring from Unisys.

Gradillas has an explanation for the decline of A.P. calculus at Garfield: Escalante and Villavicencio were not allowed to run the program they had created on their own terms. In his phrase, the teachers no longer “owned” their program. He’s speaking metaphorically, but there’s something to be said for taking him literally.

In the real world, those who provide a service can usually find a way to get it to those who want it, even if their current employer disapproves. If someone feels that he can build a better mousetrap than his employer wants to make, he can find a way to make it, market it, and perhaps put his former boss out of business. Public school teachers lack that option.

There are very few ways to compete for education dollars without being part of the government school system. If that system is inflexible, sooner or later even excellent programs will run into obstacles.

Escalante has retired to his native Bolivia. He is living in his wife’s hometown and teaching part time at the local university. He returns to the United States frequently to visit his children. When I spoke to him he was entertaining the possibility of acting as an adviser to the Bush administration. Given what he achieved, he clearly has valuable advice to give.

Whether the administration will take it is another question. We are being primed for another round of “education reform.” One-size-fits-all standardized tests are driving curricula, and top-down reforms are mandating lockstep procedures for classroom instructors. These steps might help make dismal teachers into mediocre ones, but what will they do to brilliant mavericks like Escalante?

Before passing another law or setting another policy, our reformers should take a close look at what Jaime Escalante did — and at what was done to him.

Jerry Jesness is a special education teacher in Texas’ Lower Rio Grande Valley.

 

At L.A. school, Singapore math has added value

At L.A. school, Singapore math has added value

Sum fun
By Mitchell Landsberg, Los Angeles Times Staff Writer
March 9, 2008
Here’s a little math problem:

In 2005, just 45% of the fifth-graders at Ramona Elementary School in Hollywood scored at grade level on a standardized state test. In 2006, that figure rose to 76%. What was the difference?

Sum fun
By the numbers

If you answered 31 percentage points, you are correct. You could also express it as a 69% increase.

But there is another, more intriguing answer: The difference between the two years may have been Singapore math.

At the start of the 2005-06 school year, Ramona began using textbooks developed for use in Singapore, a Southeast Asian city-state whose pupils consistently rank No. 1 in international math comparisons. Ramona’s math scores soared.

“It’s wonderful,” said Principal Susan Arcaris. “Seven out of 10 of the students in our school are proficient or better in math, and that’s pretty startling when you consider that this is an inner-city, Title 1 school.”

Ramona easily qualifies for federal Title 1 funds, which are intended to alleviate the effects of poverty. Nine of every 10 students at the school are eligible for free or reduced-price lunches. For the most part, these are the children of immigrants, the majority from Central America, some from Armenia. Nearly six in 10 students speak English as a second language.

Yet here they are, outpacing their counterparts in more affluent schools and succeeding in a math curriculum designed for students who are the very stereotype of Asian dominance in math and science.

How did that happen?

It’s a question with potentially big implications, because California recently became the first state to include the Singapore series on its list of state-approved elementary math texts. Public schools aren’t required to use the books — there are a number of other, more conventional texts on the state list — but the state will subsidize the purchase if they do. And being on the list puts an important imprimatur on the books, because California is by far the largest, most influential textbook buyer in the country.

The decision to approve the books could place California ahead of the national curve. The National Mathematics Advisory Panel, appointed by President Bush, will issue a report Thursday that is expected to endorse K-8 math reforms that, in many ways, mirror the Singapore curriculum.

The report could also signal a cease-fire in the state’s math wars, which raged between traditionalists and reformers throughout the 1990s and shook up math teachers nationwide. Fundamentalists called for a return to basics; reformers demanded a curriculum that would emphasize conceptual understanding.

Mathematicians on both sides of the divide say the Singapore curriculum teaches both. By hammering on the basics, it instills a deep understanding of key concepts, they say.

Kids — at least the kids at Ramona — seem to love it.

Ramona, which received a grant to introduce the Singapore curriculum, is one of a sprinkling of schools around the country to do so.

Not all teachers like it, and not all use it. The Singapore books aren’t easy for teachers to use without training, and some veterans are more comfortable with the curriculum they have always followed. But you can tell when you walk into a classroom using Singapore math.

“On your mark . . . get set . . . THINK!”

First-grade teacher Arpie Liparian stands in front of her class with a stopwatch. The only sound is of pencils scratching paper as the students race through the daily “sprint,” a 60-second drill that is a key part of the Singapore system. The problems at this age are simple: 2+3, 3+4, 8+2. The idea, once commonplace in math classrooms, is to practice them until they become second nature.

Critics call this “drill and kill,” but Ramona’s math coach, Robin Ramos, calls it “drill and thrill.” The children act as though it’s a game. Not everyone finishes all 30 problems in 60 seconds, and only one girl gets all the answers right, but the students are bubbling with excitement. And Liparian praises every effort.

“Give yourselves a hand, boys and girls,” she says when all the drills have been corrected. “You did a wonderful job.”

Reinforcing patterns

What isn’t obvious to a casual observer is that this drill is carefully thought out to reinforce patterns of mathematical thinking that carry through the curriculum. “These are ‘procedures with connections,’ ” Ramos said, arranged to convey sometimes subtle points. This thoughtfulness — some say brilliance — is the true hallmark of the Singapore books, advocates say.

After 10 years of studying the Singapore curriculum, Yoram Sagher, a math professor at the University of Illinois at Chicago, said he still has “very pleasant surprises and realizations” while reading the books. Sagher, who helped train Ramos and the other teachers at Ramona, said he is constantly amazed by “the gentle, clever ways that the mathematics is brought to the intuition of the students.”

The books, with the no-nonsense title “Primary Mathematics,” are published for the U.S. market by a small company in Oregon, Marshall Cavendish International. They are slim volumes, weighing a fraction of a conventional American text. They have a spare, stripped-down look, and a given page contains no material that isn’t directly related to the lesson at hand.

Standing in an empty classroom one recent morning, Ramos flipped through two sets of texts: the Singapore books and those of a conventional math series published by Harcourt. She began with the first lesson in the first chapter of first grade.

In Harcourt Math, there was a picture of eight trees. There were two circles in the sky. The instructions told the students: “There are 2 birds in all.” There were no birds on the page.

The instructions directed the students to draw little yellow disks in the circles to represent the birds.

Ramos gave a look of exasperation. Without a visual representation of birds, she said, the math is confusing and overly abstract for a 5- or 6-year-old. “The math doesn’t jump out of the page here,” she said.

The Singapore first-grade text, by contrast, could hardly have been clearer. It began with a blank rectangle and the number and word for “zero.” Below that was a rectangle with a single robot in it, and the number and word for “one.” Then a rectangle with two dolls, and the number and word for “two,” and so on.

“This page is very pictorial, but it refers to something very concrete,” Ramos said. “Something they can understand.”

Next to the pictures were dots. Beginning with the number six (represented by six pineapples), the dots were arranged in two rows, so that six was presented as one row of five dots and a second row with one dot.

Day one, first grade: the beginnings of set theory.

“This concept, right at the beginning, is the foundation for very important mathematics,” Ramos said. As it progresses, the Singapore math builds on this, often in ways that are invisible to the children.

Word problems in the early grades are always solved the same way: Draw a picture representing the problem and its solution. Then express it with numbers, and finally write it in words. “The whole concept,” Ramos says, “is concrete to pictorial to abstract.”

Another hallmark of the Singapore books is that there is little repetition. Students are expected to attain mastery of a concept and move on. Each concept builds upon the next. As a result, the books cover far fewer topics in a given year than standard American texts.

Skilled at math

Singapore is a prosperous, multicultural, multilingual nation of 4.5 million people whose fourth- and eighth-grade students have never scored lower than No. 1 in a widely accepted comparison of global math skills, the Trends in International Mathematics and Science Study. U.S. students score in the middle of the pack.

When the U.S. Department of Education commissioned a study in 2005 to find out why, it concluded, in part: “Singapore’s textbooks build deep understanding of mathematical concepts through multi-step problems and concrete illustrations that demonstrate how abstract mathematical concepts are used to solve problems from different perspectives.”

By contrast, the study said, “traditional U.S. textbooks rarely get beyond definitions and formulas, developing only students’ mechanical ability to apply mathematical concepts.”

Many eminent mathematicians agree. In fact, it is difficult to find a mathematician who likes the standard American texts or dislikes Singapore’s.

“The Singapore texts don’t make a huge deal about the concepts, but they present them in the correct and economical form,” said Roger Howe, a professor of mathematics at Yale University. “It provides the basis for a very orderly and systematic conceptual understanding of arithmetic and mathematics.”

The Singapore curriculum is not strikingly different from that used in many countries known for their math prowess, especially in Asia and Eastern Europe, math educators say. According to James Milgram, a math professor at Stanford who is one of the authors of California’s math standards, the Singapore system has its roots in math curricula developed in the former Soviet Union, whose success in math and science sent shivers through American policymakers during the Cold War.

The Soviets, Milgram said, brought together mathematicians and developmental psychologists to devise the best way to teach math to children. They did “exactly what I would have done had I been given free rein to design the math standards in California. They cut the thing down to its core.”

The Soviet curriculum was adopted by China in the mid-1950s, he said, and later made its way to Singapore, where it was rewritten and refined. The Singapore texts could easily be adapted for use in the United States because children there are taught in English.

“American textbooks are handicapped by many things,” said Hung-Hsi Wu, who has taught math at UC Berkeley for 42 years, “the most important of which is to regard mathematics as a collection of factoids to be memorized.”

One might think that school districts would be lining up to get their hands on the Singapore texts, but no one expects many to take the plunge this fall.

“Maybe in seven or eight years, but not yet,” said Wu. For now, he said he’d be surprised if the Singapore books claim 10% of the market.

In part, that may reflect the inherent conservatism of the education establishment, especially in large districts such as Los Angeles Unified, whose math curriculum specialists said in December, a month after the Singapore texts were adopted by the state, that they hadn’t even heard of them — or of the successful experiment taking place in one of their own schools.

But there is also an understandable reluctance to rush into a new curriculum before teachers are trained to use it. Complicating that, experts said, is that most American elementary school teachers — reflecting a generally math-phobic society — lack a strong foundation in the subject to begin with.

The Singapore curriculum “requires a considerable amount of math background on the part of the teachers who are teaching it,” said Milgram, “and in the elementary grades, most of our teachers aren’t capable of teaching it. . . . It isn’t that they can’t learn it; it’s just that they’ve never seen it.”

Training is key

Adding to the difficulty is that the Singapore texts are not as teacher-friendly as most American texts. “They don’t come with teachers editions, or two-page fold-outs with comments, or step-by-step instructions about how to give the lessons,” said Yale’s Howe. “Most U.S. elementary teachers don’t currently have that kind of understanding, so successful use of the Singapore books would require substantial professional development.”

Although some U.S. schools have had spectacular results using Singapore texts, others have fared less well. A study found that success in Montgomery County, Md., schools using the Singapore books was directly related to teacher training. At schools where teachers weren’t trained as well, student achievement declined.

Sagher, the Illinois professor, said that he would love to see Ramona Elementary become a training ground for L.A. Unified teachers and that Singapore math could radiate out from its Hollywood beachhead. Districtwide, only 43% of fifth-graders last year scored at grade level or above in math, 33 points below Ramona students. “If LAUSD is smart enough to do it, it will be a revolution,” he said.

mitchell.landsberg

@latimes.com