

Mathematics
"I finally got around to examining your web site and I was amazed.
What a marvelous job you have done! ...
Best regards and again, congratulations on your web site."
 Frank Allen,
Emeritus Professor of Mathematics at Elmhurst College,
National Advisor for
Mathematically Correct,
and former president, National Council of Teachers of Mathematics (NCTM)

Understanding the Battles Over Math
 Math Education: An Inconvenient Truth,
produced by
Where's the Math?, Washington State.
Meteorologist M. J. McDermott presents vivid illustrations of the problems with fuzzy math,
using specific examples from Everyday Math and TERC Investigations:
 Math Education: A University View,
produced by
Where's the Math?, Washington State.
Prof. Cliff Mass, Department of Atmospheric Sciences, University of Washington notes that
students are clearly less capable in math than 1015 years ago, leading to
the need to "dumb down" college courses accordingly. Prof. Mass lays the blame squarely on the impact of
fuzzy math in K12.
 Math with Madeline,
produced by
Where's the Math?, Washington State.
MUSTSEE VIEWING! Fifth grader Madeline shows us vivid examples of the differences between the fuzzy math program
that she used in her old school
(TERC Investigations)
and the mastery math programs
(Saxon Math and
Singapore Math) she now uses.

Kevin Killion's statement to the Chicago session of the National Math Advisory Panel, April 20, 2007.
Excerpts:
"Another weapon is to blame lousy math performance on intractable, dusty old
methods. Schools are urged to 'embrace change' and teachers are exhorted to
be 'agents of change.'
The reality couldn't be any more starkly different. Everything has already
changed.
On our Illinois Loop website, we provide extensive information about how
math is taught in Illinois school districts, from Addison to Zion. This resource is
wellused by parents in tracking what districts are doing. Here's what we've
found:
"In Chicago, some 290 schools use progressivist, constructivist math programs in
early grades. On the flip side, we have been able to identify only 5  count 'em  5
conventional CPS schools that use practiceandmastery math programs, plus
another 5 schools that are charters offering Saxon Math.
"Now the suburbs.
The Illinois Loop has collected info on the math programs used in 118 suburban
K8 districts in five collar counties. We find that progressivist, constructivist
products are the math foundation in 77 percent of those districts.
"But even that only hints at the severity of the problem. On the North Shore, or in
Lake County, it's almost impossible to find any schools with anything but
constructivist math. And across the area, we identified only 6  6!  districts 
out of 118  that make use of those math programs most recommended by
practiceandmastery reformers, such as Singapore Math or Saxon Math. So
much for the argument that parents in the suburbs already have the schools they
want.
"Here's a twist: We've all heard of the 'dance of the lemons.' Well, there is also
the 'Dance of the Math Lemons' performed by districts unhappy with their math
programs. Example: District 39 in Wilmette dumps
Math Trailblazers
and picks up
Everyday Math,
even while District 109 in Deerfield drops
Everyday Math
to take a chance on
Math Trailblazers.
Like Lois Lane who couldn't see
the truth staring her in the face, these districts stick with constructivist math and
merely substitute one program for another.
We're sure not seeing any 'agents for change' there! These districts are firmly
mired down with a philosophy they refuse to abandon."
 Here are two excellent short articles that provide a terrific introduction
to understanding the issues of math instruction. They are written by
Paul Clopton (one of the cofounders of Mathematically Correct), and are
from the August 2001 issue of Parent Power, a publication of the Center for Education Reform:

It Works for Me: An Exploration of Traditional Math, Part 1
It Works for Me: An Exploration of Traditional Math, Part 2
It Works for Me: An Exploration of Traditional Math, Part 3
by Barry Garelick, November 12, 2007.
"Anyone who has been involved in the debates surrounding math
education [has] come across the arguments that 'traditional math
doesn't work' or 'the old way of teaching math was a mass failure'."
But what do the facts say?
These three juicy articles provide background on the math texts used in the 1940s through 1960s,
leading to revealing conclusions, particularly in light of the fact that math scores
on Iowa tests steadily increased from the 40's through the mid60's, after which they began a decline.
Highly recommended!
 Here's another excellent article. This one does a good job giving specific examples
on the differences between "fuzzy math" and "traditional" methods:
U.S. Math Woes Add Up to Big Trouble
by Ken Gorrell,
Concord [NH] Monitor,
April 8, 2007. Excerpts:
"A bit of context is important. The reformers, representing the
education establishment, believe learning 'process' is more important
than memorizing core knowledge. They see selfdiscovery as more
important than getting the right answer. For them it's the journey,
not the destination.
"Traditionalists, consisting mainly of parent groups and
mathematicians, advocate teaching the traditional algorithms. They
advocate clear, concrete standards based on actually solving math
problems. The destination  getting the right answer  is important
to traditionalists.
"Two examples will help to make the difference
clear. ..."

"We sometimes lower the bar because we
want to make sure everyone gets over it."
 Chicago CPS Exec

Illinois Students Not Up To Test Mark: ACT Scores Indicator Of College Readiness
by Stephanie Banchero, Chicago Tribune, August 17, 2005. Excerpt:
"If performance on the ACT college entrance exam is any indicator,
this year's graduating Illinois high school seniors lack the academic
skills necessary to pass basic collegelevel math, science and
reading courses, according to data released Tuesday by the testing
company.
Of the 136,000 Illinois students who took the ACT, only 25 percent
posted a science composite score high enough to indicate they are
likely to succeed in a firstyear college science course. Only 38
percent met the standard in math. About half did so in reading."
 A quote from the above article, reminiscent of President Bush's complaint about
schools that fail due to their "soft bigotry of low expectations":
"...I think we are sometimes guilty of not teaching to the
rigor of those courses. ... We sometimes lower the bar because we
want to make sure everyone gets over it."
 Donald Pittman, Chicago Public Schools chief officer for high schools

Cognitive Child Abuse in Our Math Classrooms
by C. Bradley Thompson. Excerpts:
"The test results are in: America's children are flunking math. ...
As educators scramble to explain America's math meltdown ... few are willing to look at
the fundamental cause: the new, 'wholemath' method for teaching. ...
"In a typical wholemath classroom, children do multiplication not by
learning the abstract multiplication table, but by using piles of
marshmallows. They count a million birdseeds in order to understand
the concept 'million.' They measure angles by stretching rubber bands
across pegged boards. One wholemath program preposterously claims to
foster a 'conceptual understanding' of math by asking fifthgraders
the following stumper: 'If math were a color, it would be ______ ,
because ______.' Surely such exercises foster in children only
conceptual stultification  along with a bewildered sense of
frustration and disgust.
"Another wholemath program asks sixthgraders to address the
following problem: 'I've just checked out a library book that is
1,344 pages long! The book is due in three weeks. How many pages will
I need to read a day to finish the book in time?' The proper way to
solve the problem would be to use the method for long division: 1,344
divided by 21. By contrast, the wholemath approach assigns students
to a group, requires them to design their own problemsolving rules,
and urges them to guess if all else fails. In other words, children
are told that their random 'strategies' are just as good as the
logically proven principles of long division. They are taught that
the vote of the group, rather than the reasoning of the individual
mind, is the means of arriving at the truth.
"Now imagine flying on a plane designed by aeronautical engineers who
have been trained to concoct their own math schemes and to use a
'guessandcheck' method."

Math Wars: lead editorial in the Wall Street Journal, January 4, 2000.
"Reinventing math is an old tradition in this country. It has been around at least since
the 1960's, when the inimitable Tom Lehrer mocked the New Math in Berkeley cafes. Even
Beatniks understood that a method that highlights concepts at the expense of plain old
calculation would add up to trouble. ...
Today the original New Math is old hat, but many folks in the education world are hawking
yet another reform. It is known by names like "Connected Math," or
"Everyday Math." ...
Not that all members of the Academy are joining the movement. ...
200 mathematicians and scientists, including four Nobel
Prize recipients and two winners of a prestigious math prize, the Fields Medal, published
a letter in the Washington Post deploring the reforms. ...
And well they might. For programs of [this] sort ... turn out to be
horrifyingly short on basics."
 How About That: Back To Basics
Editorial, Chicago Tribune, September 25, 2006.
"In recent years, any parent who has sat at the kitchen table with a
child completing math homework has watched the pages grow fuzzier and
fuzzier. The child busily begins to attack a math problem using a
'new math' method. The parent looks on and sighs, wistful for the
days when a student could arrive at an answer in a much more concrete
and direct way.
Such is the heart of the socalled math wars, the conflict between
math traditionalists who stress the basics and those who push the
looser "constructivist" approach."

A powerful speech from a parent in an upscale suburb:
Elizabeth Gnall's statement to the National Math Advisory Panel\, September 6, 2007.
Excerpts:
"I live in the affluent public school district of Ridgewood, New
Jersey. But my district has a dirty little secret. Ridgewood Public
School district is segregated  on one side of town, elementary
schoolaged children are taught math following a logical sequencing
of topics, honoring the scholarly body of mathematics.
"In another part of town the math is not taught but instead it is left
for the children to discover and to construct. The math where for
grades beyond Kindergarten the use of scissors, glue, paperclips, and
any other object now defined as a manipulative, are deemed acceptable
and encouraged. Sadly, this is the side of town where my children
attend school. ...
"Across this nation, parents just like me, will ultimately triumph in
the math wars because it is OUR children, not the children of the
state. ... Give us a choice in math education and we would choose a
math education that is rigorous, focuses on content, is not driven by
constructivist pedagogy, emphasizes the learning of mathematical
facts, principles, and algorithms, uses the proper language and
symbolic notation of math, and defines mathematical reasoning as the
interconnections within mathematics. It is the kind of math that is
being taught in other parts of this nation, the world, and in other
parts of my town of Ridgewood, New Jersey. It is the math I believe
that will provide a solid foundation for my children so if they
desire, if they dream, to become a scientist, an architect, or like
their dad, a Wall Street finance executive, or like their mom, an
engineer, they can."

Ten Myths About Math Education, And Why You Shouldn't Believe Them
by Karen Budd, Elizabeth Carson, Barry Garelick, David Klein, R. James Milgram,
Ralph A. Raimi, Martha Schwartz, Sandra Stotsky, Vern Williams, and W. Stephen Wilson,
in association with New York City HOLD and Mathematically Correct, two education
advocacy organizations of parents, mathematicians, and K12 educators, May 4, 2005.
You can make good use of this chart to dispel these destructive claims:
 Myth #1: Only what students discover for themselves is truly learned.
 Myth #2: Children develop a deeper understanding of mathematics and a greater sense of ownership when they are expected to invent and use their own methods for performing the basic arithmetical operations, rather than study, understand and practice the standard algorithms.
 Myth #3: There are two separate and distinct ways to teach mathematics. The NCTM backed approach deepens conceptual understanding through a problem solving approach. The other teaches only arithmetic skills through drill and kill. Children don't need to spend long hours practicing and reviewing basic arithmetical operations. It's the concept that's important.
 Myth #4: The math programs based on NCTM standards are better for children with learning disabilities than other approaches.
 Myth #5: Urban teachers like using math programs based on NCTM standards.
 Myth #6: Calculator use has been shown to enhance cognitive gains in areas that include number sense, conceptual development, and visualization.
 Myth #7: The reason other countries do better on international math tests like TIMSS and PISA is that those countries select test takers only from a group of the top performers.
 Myth #8: Math concepts are best understood and mastered when presented "in context"
 Myth #9: NCTM math reform reflects the programs and practices in higher performing nations.
 Myth #10: Research shows NCTM programs are effective.

The Myths and Realities about "Fuzzy Math"
by Sandra Stotsky, July 4, 2005.
For almost two decades, mathematics education in K12 classrooms has been
driven by unsupported pedagogical theories constructed in our schools of
education and propagated by the National Council of Teachers of Mathematics (NCTM). ...
But many parents, mathematics experts, and K12 teachers of mathematics do
not share this vision. They reject the NCTM doctrine and model for mathematics
reform. The views of this diverse constituency, comprised of mathematicians,
scientists, engineers, K12 teachers of mathematics, educational researchers,
and concerned parents across our nation have been regularly marginalized by
the dominant voice of mathematics educators in our schools of education and of
NCTM officials. This constituency's expertise is often entirely absent from the
decisionmaking process.
As a member of that constituency and an advocate for authentic reform in
mathematics education, I was part of a group that decided to prepare a
pointbypoint refutation of a set of common myths spread nationally and
internationally by mathematics educators in our schools of education and NCTM officials.
These myths are often presented as facts to policy makers and the general
public. I offer this slightly revised chart for possible use by a
curriculum committee in a school or district appointed to revise its
K12 mathematics curriculum or to decide on new mathematics
textbooks, and by candidates for school boards or committees in local elections.

Traditional Math Means Never Having to Say You're Sorry
by Barry Garelick, November 30, 2007.
"The difference between traditional and presentday teaching is
striking. The emphasis is now on big concepts. These come at the
expense of learning and mastering the basics. Getting the right
answer no longer matters. In theory, it is studentcentered
inquirybased learning. In practice it has become teachercentered
omission of instruction. With the educational zeitgeist having been
planted and taken root, the development of the NCTM standards in 1989
were an extension of a long progression. To top it all off, the
reform approach to teaching math is being taught in education
schools, thus providing future teachers with 'workarounds' to those
few math textbooks that actually have merit."

The Math Wars, Conceptual Thinking, and Traditional Algorithms:
This is a more complete review of the issue,
with wellchosen examples of the differences between "traditional" math instruction and
modern fuzzy math. You'll also find useful discussion of the intentions
and underlying philosophy of fuzzy math.
 An observer (who writes under the pseudonym "John Dewey")
comments on a video (Edspresso, October 10, 2006)
shown to ed school students as an example of best practices in math:
"Another video showed a teacher with his students standing around a
table in the center of the room while he explained that day's
assignment. ...
This lesson was about parabolas, how the various constants in the
vertex form of the equation for a parabola governed its shape,
location and direction. He had them split into four groups, each
group exploring what happens when you vary one particular constant.
They were to use colored pipe cleaners to show the various parabolas
on a poster. When through, the students all convened around the
central table again and the teacher asked many questions which the
students answered, some correctly, some not. There was no 'That's
right, that's wrong', just more questions.
The teachers in both videos were extremely good at what they were
doing, which brought home an unsettling realization to me: You can be
very good at doing something that is absolutely horrible."

A Brief History of American K12 Mathematics Education in the 20th Century
by David Klein, Mathematical Cognition, 2003.
Here is a solid, wellresearched look at how math education came to its current
dismal state, after a century of unrealistic philosophizing and tinkerings with
how math is taught.
 Highly recommended!
An AMazeing Approach to Math (HTML)
or as a (formatted PDF doc)
by Barry Garelick, Education Next, Spring 2005.
This is a definitive review on fuzzy math, written by Barry Garelick, an analyst with a federal agency in Washington.
He digs solidly into fuzzy math's roots in the depths of constructivism, and the educrat
politics (and money!) behind its encroachment on our schools.
If you're involving in a math battle, get this article to help your cause.
Here are excerpts from two articles about this Garelick article:
 An Amazing Article, by Andrew Wolf, The New York Sun, March 7, 2005. Excerpt:
"Mr. Garelick was tutoring a ninthgrader who was learning geometry. ...
Mr. Garelick was horrified to learn that what he called a 'mainstay of
mathematics' was largely missing from his student's geometry text. ...
So Mr. Garelick began looking at other textbooks and found the same
truncated instruction. Students were being given a Reader's Digest
version of math."
 Math, New Math, Fuzzy Math, AntiFuzzy Math, by Bill Leonard, March 7, 2005. Excerpt:
"We collectively groan when new test scores are released
demonstrating just how poorly U.S. students perform in math compared
to their international counterparts. If you have ever asked yourself
how this sorry situation came to pass, then I encourage you to read
the article 'An AMazeing Approach to Math,' by Barry Garelick."

The Havok Wrought by Modern Math
by Dr. Charles Ormsby.
"If you are a parent of elementary school children, you've probably
seen it: elaborate makework homework assignments, cutting and pasting
extravaganzas, overly complex and roundabout procedures to add or
multiply numbers, estimation exercises that won't quit, and the use of
calculators in place of traditional arithmetic methods.
You thought: 'Of course, the educational professionals must know what
they are doing. Once my children catch on to these clever techniques,
they will develop into mathematics geniuses!'
Unfortunately, what you discover is that they never learn the core
facts and methods, their confusion grows, they lose their
selfconfidence, they decide they just can't do math, and you are
stuck paying for tutoring. Even worse, children who might have become
exceptional mathematicians, engineers, or scientists are denied their
rightful future."

2+2=5: Fuzzy Math Invades Wisconsin Schools (PDF file) by Leah Vukmir.
This article is highly recommended, both for those new to the issue
and those who have been battling for years. It provides a thorough and
passionate review of what the math controversy is all about. (Note: Although the title refers
to Wisconsin, the material covered would be useful nationwide.)
When written, the author was the organizer of an extremely effective group of parents
and teachers woprking for academic reform. She went on to become
a visiting fellow with the Wisconsin Policy
Research Institute, and she is now a State Representative.

Math Fluency by Mary Damer.
Mary writes about four stages of learning and how this view applies
to learning and mastering mathematics.
 Is This Math Program Proven? by Mary Damer.
 Mary gives tips on how to dissect the publisher's presentation and sales material,
and to determine whether evidence presented on behalf of a math program
has any validity or merit.

How To Respond When Your School Announces a NewNew Math Program by Kevin Killion.
What do you say? How do you respond when your school tells you that your
child's math program is going to be replaced? What is your reaction when
the replacement's main advantages are a "Tokyo by Night" layout,
fuzzyheaded but politically correct examples, oddball algorithms
and methods (or no methods at all), and a big emphasis on writing
essays and playing games?
Here are some suggested responses!

NCTM and "Problem Solving"
by Charles E. Breiling, December 30, 2003.
"One thing the NCTM (National Council of Teachers of Mathematics) is
good for is revisionist history. The idea that we used to teach
students the multiplication table, and then never used that
information in solving math problems is preposterous. ... In the real world, if
you teach basic skills (facility with number facts, fractions,
decimals, percents, algebra, etc.) and have students use these facts
to solve problems (without using calculators) then what you end up
with is really good problem solvers!
But in the NCTM universe, if you depend on a calculator for your
number facts, you can just jump in with your "problem solving."
Sounds reasonable enough, but what you end up with is a student who
sees "A man has 5 trucks, and each truck holds 10 cases"immediately
adding 5+10 (on the calculator, natch) for the answer of 15 cases.
Students who don't know beans about number facts won't be able to
solve problems, no matter how much you focus on problem solving!
This isn't a case of putting the cart before the horse, this is a
case of removing the horse entirely."

How NOT To Teach Math by Matthew Clavel, City Journal (NYC), Winter 2003.
In New York City, a Whole Language reading program and the very
fuzzy "Everyday Math" (a.k.a. "Chicago math") program have been mandated
for use systemwide by central administrators. Here is one NYC teacher's
view on what fuzzy math has done to his classroom and school.

The Math Wars by David Ross, Ph.D.
This is a very lucid and compelling essay that does a balanced and fair job presenting
the arguments of both reformers and "traditionalists". Ross
(a mathematician at Kodak Research Labs) then firmly concludes
in favor of the traditional point of view. An especially novel aspect
of this report is the author's dissection of just what is meant by
"conceptual thinking." Using examples in math and from elsewhere,
Ross concludes that the basics are absolutely necessary, so that
students can eventually move (as he has titled the last section)
"From Addition to Wonder".

Independent Analysis of Mathematics Textbooks (PDF)
by Chris Patterson, January 1999.
This is an extremely helpful review of the "math wars"
controversy. This paper, prepared by the Texas Public Policy Foundation
and Education Connection of Texas,
includes background on the opposing viewpoints in math instruction,
gives examples of each, and shows how these views on math
affect math textbooks. It concludes with a clear and attractively formatted summary
of detailed curriculum reports conducted by Mathematically Correct

Uncivil War: A Bloodless Account of a Bitter Battle, by Ralph Raimi, Education Next, Spring 2004.
The failure of the education establishment to support proven, effective
math instruction has far more to do with raw political battles
than it has to do with scholarly research. This article looks at the progress of
improved math instruction in California and the efforts of reformers
and the resistance of education bureaucrats. Prof. Raimi also provides
an
extended version (PDF) of the same article, which adds more details than could
be accommodated in the printed version.

How Did It Ever Come to This? by Ralph Raimi, notes for a talk to
meeting of National Association of Scholars, New York, New York, May 22, 2004.
Outlines the genesis of fuzzy math, and gives examples of how fuzzy math precepts
handicap mathematical learning.

The Math Wars  1960's Revisited by Erica Carle, October 13, 2003.
Debates over the merit of teaching basic math facts reminded this author
of a memo exchange over the same topic  40 years ago.

"Applications and Misapplications of Cognitive Psychology to Mathematics Education",
by John R. Anderson, Lynne M. Reder and Herbert A. Simon,
Department of Psychology, Carnegie Mellon University, Excerpt:
"...Some of the central educational recommendations of these [educational] movements
have questionable psychological foundations. We wish to compare these
recommendations with current empirical knowledge about effective and
ineffective ways to facilitate learning in mathematics and to reach
some conclusions about what are the effective ways. A number of the
claims that have been advanced as insights from cognitive psychology
are at best highly controversial and at worst directly contradict
known research findings. As a consequence, some of the prescriptions
for educational reform based on these claims are bound to lead to
inferior educational outcomes and to block alternative methods for
improvement that are superior."

We have 25 boxes of paper clips, 100 per box. Take away 50.
How many are left? Here's the fuzzy math way to do it in third grade  and get the wrong answer.
(Click to enlarge)

 What is the role of "learning the basics" in mathematical operations
and fundamental algorithms? How do repeated practice and mastery of computation lead to
"higherorder thinking" later? Here's a wonderful essay that gets to the heart
of these questions!
In Defense of "Mindless Rote"
by Ethan Akin, professor of mathematics, City College of New York.

A Quarter Century Of U.S. 'Math Wars' And Political Partisanship
(preprint)
by David Klein, California State University,
Journal of the British Society for the History of Mathematics, Volume 22, Issue 1, p. 2233 (2007).
Is fuzzy math a leftwing plot? Is traditional math a farright obsession?
In this interesting paper, the author laments,
"Why did disagreements about school mathematics books in the US diverge according to left and right politics? ...
In the course of the math wars, parents of school children and
mathematicians who objected to the dearth of content were dismissed
as right wing, but there is nothing inherently left wing about the
NCTM aligned mathematics programs. ... Progressive math is a
purely capitalist phenomenon. Indeed, one of the promotional themes
of the NCTM was to prepare students for the needs of business.
Ultimately, the injection of left and right ideologies into
mathematics education controversies is counterproductive. The math
wars are unlikely to end until programs espoused by progressives
incorporate the intellectual content demanded by parents of school
children and mathematicians."

Newage Math Doesn't Add Up
by Bruce Ramsey, editorial columnist, Seattle Times, April 22, 2007.
"It's called reform math, discovery math, constructivist math, fuzzy
math. I think of it as newage math, and believe it is one reason why
last year nearly half the 10thgraders in Washington public schools
failed the mathematics portion of the highschool graduation test. It
is also one reason American kids do so poorly when measured against
kids from Europe and East Asia. ...
"Newage math ... came packaged
with a garden basket of fragrant thoughts. ... It tends to introduce topics
in a roundabout way that aims for a eureka moment. That is the
'discovery' part. It introduces many subjects early, focusing on
concepts rather than calculation. That is the 'constructivist' part.
It sometimes wants the student to estimate an answer rather than find
the right one. That is the 'fuzzy' part. It demands written
explanations of how an answer was arrived at, often in 'math
journals.' ...
Newage math uses games, colored blocks, dice, poker chips and other
manipulatives. It requires working in groups. 'The idea is that if
you let them struggle and come up with their own solutions, they'll
learn it better,' ...
None of these things is necessarily bad. ... But there are drawbacks. ...
"[A teacher] at Ballard High, says, 'Supposedly, reform math is heavier in concepts
but weaker in skills. But in my experience, kids are weaker in both.'
He says the weakness is most noticeable in 'B' and 'C' students. ...
And after high school? At community colleges, half the students take
remedial math. At the University of Washington, [a professor] says,
'I saw a profound drop in math skills
starting in the mid'90s.' Newage math, he says, has created
'a whole generation of students who can't do fractions.'"

Calculating the Effects of "Discovery" Math
by Bruce Ramsey, editorial columnist, Seattle Times, May 16, 2007.
"A mother in Everett
wrote, 'I discovered this past year that my eighthgrader is
calculatordependent ... The math skills she lacks stem from the fact
that she never learned her basic math facts. She doesn't know how to
do long division. She relies on her calculator for simple math
problems.'
"Says a Bellevue mom with a son, 15: 'Students are told to create their
own algorithms to solve addition and subtraction problems, and these
algorithms are frequently incomplete and unreliable. [Students] are
presented with a little probability, introduced to matrices, presented
with a smattering of this and that, but never achieve mastery of a
topic. They are not taught long division. They are taught to use
calculators to do the most simple problems.'
"Writes a Redmond mom: 'Our fifthgrader has not been taught how to
multiply doubledigit numbers without a calculator, or what the heck
to do with long division.'
"A Shoreline dad helping his seventhgrade daughter had forgotten the
rule for solving a math problem. He discovered that the rule wasn't in
the book. The kids were supposed to figure it out themselves. Math, he
grumbled, was being taught 'like philosophy, with no set rules and
right answers.' ...
"A Boeing engineer says he is 'constantly amazed by the gap in math
skills between our junior American engineers and those educated in any
other country,' especially those with the British system of education.
He and his wife are teaching their two kids at home, using Singapore Math.
"Another email was from a MarysvillePilchuck High School graduate who
scored so low on a math placement test that it shocked him.
'I went back to some older textbooks my mother had,' he writes. He
worked with them, retook the test, and placed himself three levels
higher. He is now graduating from the University of Washington,
though, he writes, with 'no thanks to my highschool math.'"

The Demise of Basic Surveying Mathematics
by Richard L. Elgin, Ph.D., American Surveyor, May 2007.
"The entering students' knowledge of the basic subjects of algebra, trigonometry and geometry
has sunk to such a low level in the past five to ten years, that for
all practical purposes I can say they have virtually none when it
comes to being prepared to attack surveying. Discussing such pre
surveying topics as triangle solutions, orthogonal vectors calculation
(i.e., latitudes and departures), even doing something simple as
recognizing similar triangles when trying to derive horizontal curve
equations just draws blank stares. What's the problem? I believe we
can lay most of the blame at the feet of high school math curricula."
 How should we approach math education for children with disabilities? This paper tackles
this question headon:
Educational Aspects of Mathematics Disabilities
by Susan Peterson Miller and Cecil D. Mercer, Journal of Learning Disabilities, January/February 1997.
Excerpt:
"Numerous educators have expressed concern regarding the application of the [NCTM] Standards
to students with disabilities (Carnine, 1992; Hofmeister, 1993; Hutchinson, 1993;
Mercer, Harris, & Miller, 1993; Rivera, 1993).
Among the concerns are the lack of references to students with disabilities in the [NCTM] Standards document,
lack of research related to the [NCTM] Standards, and overall vagueness of the document.
These issues need to be addressed if we are to avoid another failed reform movement,
with students paying the greatest price."

Guide to Mathematics and Mathematicians on The Simpsons:
Here's an astonishing collection of every morsel of math that has even been on the show,
compiled by Dr. Andrew Nestler and Dr. Sarah J. Greenwald.
It includes some terrific quotations from the Simpsons, including plenty from the legendary
"Girls Just Want to Have Sums" episode, as well as some screen captures.
The website even includes some lovinglyprepared classroom
worksheets on math topics mentioned on the show!
Practice Leads to Insight and Mastery

Discovery Learning in Math: Exercises Versus Problems
by Barry Garelick, Nonpartisan Education Review, Vol.5, No.2, 2009.
Ed school theorists promote the notion that traditional math courses
provide merely mechanical and
algorithmic approaches that do not lead to "off the script" thinking.
This article takes a close look at how
socalled "exercises" do indeed lead to discovery and to unscriptlike thinking.
Excerpt:
"Whether in driving, math, or any other undertaking that requires knowledge and skill, the
more expertise one accumulates, the more one can depart from the script and successfully
take on novel problems. It's essential that at each step, students have the tools, guidance, and
opportunities to practice what they learn. It is also essential that problems be well posed.
Openended, vague, and/or illposed problems do not lend themselves to any particular
mathematical approach or solution, nor do they generalize to other, future problems. As a
result, the challenge is in figuring out what they mean  not in figuring out the math. Wellposed
problems that push students to apply their knowledge to novel situations would do
much more to develop their mathematical thinking."
"Students given welldefined problems that draw upon prior knowledge, as described in this
article, are doing much more than simply memorizing algorithmic procedures. They are
developing the procedural fluency and understanding that are so essential to mathematics;
and they are developing the habits of mind that will continue to serve them well in more
advanced, college level mathematics courses. Poorlyposed problems with multiple 'right'
answers turn mathematics into a frustrating guessing game. Similarly, problems for which
students are expected to discover what they need to know in the process of solving it do little
more than confuse. But wellposed problems that lead students in manageable steps not only
provide them the confidence and ability to succeed in math, they also reveal the logical,
hierarchical nature of this powerful and rewarding discipline."

Math Discoveries Catch Kids Unawares
by B. Bower, Science News, January 2, 1999.
Many of those who have careers that depend on math generally feel they learned it not by talking
about "ways" of solving problems, but rather by doing math, with lots and lots of practice.
This article from Science News reports on an interesting study that suggests exactly that!
Three amazing conclusions from this study:
 kids deduced an important mathematical algorithm simply from doing computation problems,
 nearly all of the kids in the study eventually did this, and
 most of the kids were not able to explain in words what they were doing,
thus suggesting that being forced to convert mathematical understanding into
words was a skill quite distinct from mathematical competency.
Here are some excerpts:
"Many educators and scientists assume that conscious knowledge is the
engine that drives learning. A new study suggests instead that, at
least among gradeschoolers, unconscious problemsolving insights
often set the stage for academic advances.
Secondgraders who practice solving inversion problems  such as
8+1010 = 8  start out by computing the answers but frequently turn
to a more efficient strategy unconsciously. ...
However, after becoming aware of the shortcut, kids employ it only
part of the time, returning at other times to more timeconsuming
calculations. In the long run, the child's nurturing of an array of
problemsolving tactics allows for adjustments in tougher math
challenges. ... 'This study shows that conceptual insights emerge
unconsciously during practice,' remarks psychologist David C. Geary
of the University of Missouri in Columbia."

How Can Learning Facts Make Thinking More Enjoyable  and More Effective?
by Daniel T. Willingham, American Educator, American Federation of Teachers, Spring 2009.
"Data from the last 30 years lead to a conclusion that is not
scientifically challengeable: thinking well requires knowing facts,
and that's true not simply because you need something to think about.
The very processes that teachers care about most  critical thinking
processes like reasoning and problem solving  are intimately
intertwined with factual knowledge that is in longterm memory (not
just in the environment). ...
"Take two algebra students  one is still a
little shaky on the distributive property, whereas the other knows it cold. When
the first student is trying to solve a problem and sees a(b + c), he's unsure
whether that's the same as ab + c or b + ac or ab + ac. So he stops working on
the problem, and substitutes small numbers into a(b + c) to be sure that he's got
it right. The second student recognizes a(b + c), and doesn't need to stop and
occupy space in working memory with this subcomponent of the problem. Clearly,
the second student is more likely to successfully complete the problem."

"Most mathematics textbooks follow precisely the
approach that our studies find so ineffective"

Temporal Spacing and Learning
by Hal Pashler, Doug Rohrer, and Nicholas J. Cepeda,
The Observer, Association for Psychological Science (APS), March 2006.
"Studies going back a century and more have found that spacing
learning episodes across time sometimes enhances memory. The
socalled spacing effect is the topic of hundreds of articles, and
one might assume that we know all we need to know about it. However,
the subtitle of an article on spacing effects that Frank Dempster
published in American Psychologist in 1988  "A case study in the
failure to apply the results of psychological research"  remains
appropriate now. Whether one looks at classrooms, instructional
design texts, or language learning software, there is little sign
that people are paying attention to temporal spacing of learning.
"[O]ur team ... has been teaching students
abstract mathematics skills ... In a study ..., students learned to solve a type of
permutation problem, and then worked two sets of practice problems.
Oneweek spacing separating the practice sets drastically improved
final test performance (which involved problems not previously
encountered). In fact, when the two practice sets were backtoback,
final performance was scarcely better than if the second study
session was deleted altogether. This fits with other research from
our team showing that benefits of overlearning decline sharply with
time ...
"Interestingly, most mathematics textbooks follow precisely the
approach that our studies find so ineffective: a brief lesson on a
topic is followed by a practice set containing virtually every
problem in the book relating to this topic. Far more useful, we
suspect, is to intersperse problems related to older topics covered
over past weeks and months."
 "The ideal textbook also has a tremendous number of practice problems because
practice, practice and more practice is demanded from each student. Finding
enough practice problems has always been difficult, thus I am currently
developing my own texts and practice workbooks for class and homework use that
are consistent with the lesson plans of the program."
 legendary math teacher Jaime Escalante
Fuzzy Math Impedes Learning
"This year's graduating Illinois high school seniors lack the academic
skills necessary to pass basic collegelevel math, science and
reading courses ...
"Of the 136,000 Illinois students who took the ACT ... only 38
percent met the standard in math."
Chicago Tribune August 17, 2005


An Illusory Math Reform; Let's Go To The Videotape
by Linda Seebach, Rocky Mountain News, August 7, 2004.
Excerpt:
"American children come off badly in international comparisons of
mathematics performance, and they do worse the longer they're in
school. One such comparison, the Third International Mathematics and
Science Study, tested more than 500,000 children in 41 countries,
starting in 1995. As part of the study, researchers videotaped more
than 200 eighthgrade math lessons. These lessons have been studied
intensively in an effort to figure out why Japanese students do so
well in math while American students do so badly.
[But] Alan Siegel, a professor of computer science at New York
University, ... believes that many of the [published articles about
the] TIMSS studies contain 'serious errors and misunderstandings.'
[The videotapes reveal] teaching in the traditional mode, beautifully
designed and superbly executed, but nothing like the parody of
instruction that goes by the term 'discovery learning' in mathreform
circles in the United States.
The videotape shows, Siegel says, that 'a master teacher can present
every step of a solution without divulging the answer, and can, by so
doing, help students learn to think deeply. In such circumstances,
the notion that students might have discovered the ideas on their own
becomes an enticing mix of illusion intertwined with threads of
truth.' Illusion prevails in far too many American classrooms."

Trends in Math Achievement: The Importance of Basic Skills,
presentation by Tom Loveless, Senior Fellow, Governance Studies,
Brookings Institution, February 6, 2003. Excerpts:
"Take a closer look at the scores for nine year olds. These skills
comprise the basic arithmetic that all fourth graders are expected to
master  addition, subtraction, multiplication, and division of whole numbers.
All four areas reversed direction in the 1990s, turning solid gains that
were made in the 1980s into losses. ...
"A similar concern can be raised about the performance of thirteen and
seventeen year olds. Their level of proficiency on computation skills
remains unacceptably low. Look closely at fractions. Proficiency with
fractions is critical in preparation for algebra. In 1999, only about
half of thirteen and seventeen year olds could compute accurately with
fractions on the NAEP. Students who leave eighth grade not knowing how
to compute with fractions enter high school as remedial matmanipuh students.
Students who leave high school lacking proficiency with fractions are
inadequately prepared for college mathematics.
On the most recent trend NAEP, both age groups were less proficient at
computing with fractions than in 1982, twenty years ago."
Lattice Multiplication
 
 "Lattice multiplication" 

NewMath Multiplies
by Linda Schrock Taylor.
"Yes, NewMath is multiplying, but I am sorry to
report that too many children are not learning to
multiply with NewMath. ... Multiplication is not all
that difficult if one learns the multiplication tables
and the logical, precise algorithm for the process. One day I was
teaching traditional multiplication when one of the
special education students wanted to show me the
process she had been taught. Her problem even shocked
me, and luckily I had my camera with me. This
illustration should help the unaware to understand why so many
children in special education, as well as most other
children, are coming to believe that math is an alien
life form. It is no wonder that, when such foolishness
is passed off as an intelligent math procedure, math
scores are dropping like stones, while confusion is rising
to new heights. It is no wonder that our students grow up: seeing
themselves as stupid, hating math, and actually
mathematically incompetent!"
 Posting on an education discussion board:
"Towards the end of 3rd grade ...
I wrote a letter to my son's teacher/principal telling them that
unless they could provide me with compelling evidence that using the
lattice method of multiplication was helpful (yes, Everyday Math),
I expected my son to do all of his multiplication problems using
the standard algorithm so that he would get very good at one method,
rather than notsogood at four methods. Shortly thereafter, he came
home with a math assignment. The class had been given a worksheet
and told to use the lattice method to solve the problems, but Eric's
teacher told him he was supposed to use the standard algorithm. I
asked him what he thought about that. His response? 'Mom, it took
the other kids three times as long to get their work done.' Duh."
 What Is 5536 Divided By 82?:
Compare the standard method versus the "method" used in Everyday Math:
Do "Manipulatives" Make Math Harder?

Don't Be Manipulated,
Center for Education Reform.
"The use of "manipulatives" has become a buzzword in education
circles. The concept refers to kindergarten and elementary students'
use of concrete objects  anything from blocks or magnetic letters to
complete systems specially designed for use in the classroom  for,
literally, handson learning of math and language concepts.
"However, a recent study brings into question their effectiveness  in
this specific case, the use of a particular set of manipulatives
usually did not transfer into faster or greater proficiency in the
symbolic, written worlds of math and language.
"The lesson here for parents? Just because something's "hot" doesn't
mean it's helpful. Check out the science behind the curricular
buzzwords. Get specifics on what classroom methods are being used, if
they require special teacher training (and whether your teacher has
it), and whether their effectiveness has ever been independently
assessed.
And finally, make sure your school keeps the focus on results, and
not just process."

Mindful of Symbols: Educational Ramifications
by Judy S. DeLoache, Scientific American, August 2005.
(
Full article here.) Excerpt:
"Teachers in preschool and elementary school classrooms around the
world use 'manipulatives'  blocks, rods and other objects designed to
represent numerical quantity. The idea is that these concrete objects
help children appreciate abstract mathematical principles. But if
children do not understand the relation between the objects and what
they represent, the use of manipulatives could be counterproductive.
And some research does suggest that children often have problems
understanding and using manipulatives.
"Meredith Amaya of Northwestern University, Uttal and I are now
testing the effect of experience with symbolic objects on young
children's learning about letters and numbers. Using blocks designed
to help teach math to young children, we taught six and
sevenyearolds to do subtraction problems that require borrowing (a
form of problem that often gives young children difficulty). We
taught a comparison group to do the same but using pencil and paper.
Both groups learned to solve the problems equally wellbut the group
using the blocks took three times as long to do so. A girl who used
the blocks offered us some advice after the study: 'Have you ever
thought of teaching kids to do these with paper and pencil? It's a
lot easier.'"

Learning From Symbolic Objects
by David H. Uttal and Judy S. DeLoache, the Observer,
Association for Psychological Science (APS), May 2006.
Note of local interest: Uttal is an associate professor of psychology at Northwestern University.
"In the classroom, teachers sometimes use more formal manipulative
systems composed of concrete symbolic objects, such as Cuisenaire
Rods or DigiBlocks, that have been explicitly designed to help young
children learn mathematics. ... Based on the writings of scholars such as
Piaget, Bruner, and Montessori, educators have suggested that young
children learn best through the use of highly concrete objects.
However, our prior research on a variety of symbol systems (e.g.,
scale models, pictures, and maps) leads us to think twice about the
value of having young children play with objects that are intended to
be used as symbols. ...
letter and number toys as representations may have just the opposite effect
than what is intended: making children focus more on them as objects and less on what they stand for.
"... we investigated the effectiveness of concrete symbolic objects, known
as manipulatives, in helping young elementaryschool children learn
the procedures associated with twodigit subtraction. We taught
children using either using the traditional written method or a
commercially available manipulatives set, per the manufactures
instructions. ... We found that children initially performed equally well in both
training conditions. However, those children who learned with the
manipulatives had trouble transferring knowledge to written versions
of the math problems; they did not use what they had learned using
the manipulatives to solve written versions of the same or similar
problems. Moreover, learning with the manipulatives took almost three
times as long as learning with the written method. This result does
not mean that manipulatives are never useful, but it does challenge
the typical assumption that they are more effective than other
teaching tools in all contexts."
 Posting on an education discussion board:
"My daughter was taught to use the Cuisinaire blocks, lining up
yellows against greens and all that. ...
In later years, when she was an adult, I mentioned the Cuisinaire
blocks and asked her if they had been of any help. She said she had no
idea, at the time, that they were trying to teach her arithmetic. It was
just another 'activity.'"
Spiraling ("The Death Spiral")
Math Anxiety
Parents with a child with severe math anxiety should find these articles
extremely interesting!

Math Fears Subtract From Memory, Learning
by Bruce Bower, Science News, June 30, 2001, Vol. 159, No. 26.
Excerpt:
"Some scientists have
theorized that kids having little math aptitude in the first place
justifiably dread grappling with numbers.
That conclusion doesn't add up ... according
to a study [that concludes] people's intrusive worries about math
temporarily disrupt mental processes needed for doing arithmetic
and drag down math competence. ...
"Math anxiety exerts this effect by making it difficult to
hold new information in mind while simultaneously
manipulating it ... Psychologists regard
this capacity, known as working memory, as crucial for
dealing with numbers.
'Math anxiety soaks up workingmemory resources and makes it
harder to learn mathematics, probably beginning in middle
school' [a researcher says]."

Researchers: Math Anxiety Saps Working Memory Needed To Do Math,
Reuters, February 20, 2007.
"Worrying about how you'll perform on a math test may actually
contribute to a lower test score, U.S. researchers said on Saturday.
Math anxiety  feelings of dread and fear and avoiding math  can
sap the brain's limited amount of working capacity, a resource needed
to compute difficult math problems, said Mark Ashcroft, a
psychologist at the University of Nevada Las Vegas who studies the
problem. 'It turns out that math anxiety occupies a person's working memory,'
said Ashcroft ... Worrying about math takes up a large chunk of a person's working
memory stores ... spelling disaster for the anxious student who
is taking a highstakes test."
Calculators

Why Johnny Can't Add Without a Calculator
by Konstantin Kakaes,
Slate, June 25, 2012.
"Technology is doing to math education what industrial agriculture did to
food: making it efficient, monotonous, and lowquality."

Calculating the Cost of Calculators
by Lance T. Izumi, Capital Ideas, December 21, 2000.
"A September 2000 Brookings Institution study found that calculator use
decreases student math achievement. Analyzing national test data,
Brookings concluded that students who used calculators every day
scored lower than students who used the devices less frequently.
Given this finding, it is disturbing that Brookings also found that
while only 27 percent of white students used calculators daily, half
of AfricanAmerican students made daily use of calculators.
Yet, despite such evidence, university schools of education, which
place so much emphasis on the learning 'process,' actively promote
the use of calculators, devices that eliminate the process of
learning math. For instance, a book on math teaching methods required
at San Francisco State University tells future teachers that 'there's
no place for requiring students to practice tedious calculations that
are more efficiently and accurately done by using calculators.' ...
The evidence shows that calculator use and other trendy teaching methods
harm rather than help students. These failed methods end up cheating
children out of the basic knowledge and skills they will need to succeed.
That is both a fraud and a tragedy."
 TI Celebrates 40th Anniversary of Calculator
by Andrew D. Smith, Dallas Morning News, September 30, 2007.
Excerpt:
"... they distressed math lovers, who said calculators diminished
understanding. Math teachers, for example, spent years resisting
calculators.
Older engineers took a similarly skeptical view. [TI engineer Jerry] Merryman, who
kept using his old slide rule long after he invented the calculator,
says that engineers who learned their trade postcalculator have no
'feel for numbers.'
TI eventually found itself in an odd place. It was a math and science
company that made a fortune selling students a product that some said
hurt their understanding of math and science."

K12 Calculator Usage and College Grades (PDF)
by W. Stephen Wilson and Daniel Q. Naiman,
Educational Studies in Mathematics 56: 119122, 2004.
This study concludes that students in the math courses at Johns
Hopkins University who had been encouraged to use calculators in K12 had lower
grades than those who weren't.
 Just Say No To Calculators
by David Klein, professor of mathematics at California State University,
American Teacher, American Federation of Teachers (AFT), March 2001.
"One of the most debilitating fads to sweep American public schools in
the last decade has been the heavy use of calculators, especially in
elementary schools. According to the Third International Mathematics
and Science Study, or TIMSS, use of calculators in U.S. fourthgrade
mathematics classes is about twice the international average. In six
of the seven topscoring nations in the study, teachers of 85 percent
or more of the students report that students never use calculators in
class. ...
Even universities are forced to run remedial math classes at unprecedented levels, including classes in arithmetic for entering freshmen.
This intolerable state of affairs can be laid at the doorstep of the uncritical use of calculators in elementary schools."

"... doomed to solving only
trivial mathematical problems ..."

Computation Skills, Calculators, and Achievement Gaps: An Analysis of NAEP Items
by Tom Loveless, Director, Brown Center on Education Policy, Senior Fellow, Governance Studies,
April 15, 2004.
This valuable paper examines national trends in computation skills,
investigates whether allowing calculators on NAEP items produces
significantly different results compared to not allowing calculators,
and analyzes the impact of allowing calculators on the performance
gaps among black, white, and Hispanic students. It concludes:
"If students are only able to compute accurately with calculators  or if
their computational skills are so weak that only the simplest of
calculations can be made  then students are doomed to solving only
trivial mathematical problems."

PaperandPencil Math by Richard H. Escobales Jr., Canisius College,
Notices of the AMS, American Mathematical Society, August 2004 
a mathematician takes on those who give lip service to the learning of algorithms
while in reality favoring calculator use over the steps necessary to actually attain competency.
"Mastery of addition and the other algorithms of basic arithmetic act as a flashlight,
allowing the young student to move freely about in the world of numbers and basic
numeric operations. Without such mastery a young student is condemned to move
about blindly in this intriguing unknown world of numbers."

Calculator Dependence
by William Kohl, June 7, 2006. The author begins by describing a conversation he has with
a student he is tutoring. An algebra problem asks, in the equation
y=x^{2}+5, if the constant 5 is changed to 1, how does the plotted curve change?
"I said, 'What would you do to find the answer?' He said, 'I have to
get my calculator.' I said, 'Why?' He said, 'I need it to work the
problem.' I said, 'Couldn't we just think about the problem first?'"
The author concludes, "dependence on the calculator has short circuited the learning
of math and the development of analytical skills ...
younger people are not comfortable with numbers."

"The calculator subtly undermines the whole math curriculum"

Kick Calculators out of Class by David Gelernter, professor of
computer science at Yale University, New York Post,
May 21, 1998. "They should be banned from American
elementary schools. ... The calculator subtly undermines the whole math curriculum."

Trash the Calculator, It's Back to Basics in Britain
by Alexander MacLeod, Christian Science Monitor, January 26, 1998

Of Culture, Calculators, Math Anxiety And How We Learn,
UniSci News, May 14, 2001:
"'Better performance on the complex arithmetic [problems in a study] was associated with lower
reported calculator use in elementary and secondary school,' says [Jamie] Campbell
[of the University of Saskatchewan].
Complex arithmetic places special demands on shortterm memory skills
that simple arithmetic usually does not, because complex arithmetic
involves operations such as carrying, borrowing and placekeeping.
'This is demanding mental juggling for most people's shortterm working
memory processes,' says Campbell. 'Using a calculator might restrict the
level of expertise achieved with respect to shortterm memory
skills for complex arithmetic.'"

Calculators Are Like Bicycles
by Charles E. Breiling, April 21, 2004.
"We could go on for days about what we really think of calculators in
K12 math classes (we think they're great for science classes,
especially stochiometry in Chemistry), but suffice to say it just
might be one of the signs of the Apocalypse.
Personally, we think calculators ... are great, and they sure do save a
lot of time.
But that's not the point behind teaching children arithmetic in
school.
The point of doing paperandpencil arithmetic (for example, finding
the product of 47 and 9) isn't to actually find the product (like we
didn't know what the answer was, and that's why we have kids do these
problems). The whole purpose of this exercise is to practice a skill
accurately to the point of automaticity."
 A
discussion about calculators on Joanne Jacobs' website provided some fascinating comments.
Here are some excerpts:
 "I've taught calculus for a couple summers ...
I was frustrated beyond belief that students couldn't give a rough sketch of a function
without those darn graphing calculators... I actually forbade their use in my classroom and on tests ...
but then I was further shocked when I heard the A.P. Calculus exam requires these darn graphing calculators ...
when did they do that? WHY did they do that???"
 "The reason the students are fooling around with calculators instead of learning calculus, of course,
is that the students *can't* learn calculus because they never mastered algebra."
 "my daughter was utterly baffled by some of her Algebra II homework before I shoved the
(required) graphing calculator aside and showed her how to do the work the oldfashioned way.
She's since shown some other students in her class who were similarly stumped."
 "By analogy, carpentry is not about operating a nail gun or a power saw. Carpentry is knowing
how to cut the wood to fit and where to put the nails. A skilled carpenter can produce more
and better work with power tools, but without the underlying skill, he'll only make more sawdust and noise."

"speed up the process of not learning anything"

Calculators in High School Linear Algebra, by Prof. Ralph Raimi.
The author describes this article as telling
"how recent NCTMapproved advances in mechanized pedagogy can speed up
the process of not learning anything."

Calculated Controversy: Do HighTech Calculators Take the Challenge Out of Learning Math?
Excerpt:
"Among the leading opponents is HungHsi Wu, a math professor at the University of California,
Berkeley, who worries that calculators may deprive students of a critical tool.
'I want them to learn how to think,' he says. 'That's the greatest virtue in having a
good mathematics education. ... At some point, there's no point in using a calculator,'
he reasons, 'because if you can't think anyway, then what's the point of using a calculator?'"

A Horizon without Calculators, August 31, 1996.
Excerpt:
"There has been much concern about student use of calculators. Many
parents feel that their children exercise their fingers too much and
their brains too little by the overuse of calculators. Examples
include algebra students using calculators to solve 300/3 or 63/9.
These should be immediately solved by mental math, virtually without
stopping to think about it. When students reach for calculators to do
simple problems something is wrong. Such behavior is learned by
having calculators present at all times, a policy that is common in
many 'reform' math classes."
 Ditch the Calculators:
Letting Children Punch Numbers Into a Machine Does Not Add Up To Learning Math
by Diane Hunsaker, Newsweek,
November 3, 1997.
"Some teachers argue that calculators let students concentrate on how
to solve problems instead of getting bogged down with tedious
computations. ... Some of my elementaryschool children look at a
word problem and instantly guess that adding is the correct approach.
When I suggest that they solve the problem this way without a
calculator, they usually pause and think before continuing. A student
is much more likely to minimize his work by reflecting on the problem
first if he doesn't have a calculator in his hand. ...
A student who learns to manipulate numbers mentally can focus on how
to attack a problem and then complete the actual computations easily.
He will also have a much better idea of what the answer should be,
since experience has taught him 'number sense,' or the relationship
between numbers. A student who has grown up with a calculator will
struggle with both strategies and computations."

The Great Calculator Debate:
Concerns About Calculator Use in Elementary Schools
by Thomas W. Cowdery

Calculators in Class: Freedom From Scratch Paper Or 'Crutch'?
(also
here)
by Mark Clayton, Christian Science Monitor, May 23, 2000. Excerpts:
"Critics ... say calculators are overused in US middle and high
schools. They warn that a wave of 'new new math' programs that employ
calculators much more than traditional approaches are entering grade
schools, threatening basic math skills.
David Klein, a math professor at California State University in
Northridge, says calculators should 'not be used at all in grades
K5, and only sparingly in higher grades.' That's not where America's
schools are headed, however. ...
"And as calculators become cheaper and
more powerful, even some college students question their impact on learning. 'I
feel as though three years of math at high school were lost,' says Amir Emami, a
freshman at Kalamazoo College. Even though he graduated with a 3.4 grade point
average, he says he has a weak understanding of math. 'The answers [in high
school] were written paragraphs, not equations or number crunching. You learn to
depend on our TI82 calculator.'
"Of course, math performance is tied to many factors, Klein notes, but the
highestperforming countries on international math tests used calculators less.
At the eighthgrade level, students from three of the top five performing nations
in math (Japan, Korea, Belgium) rarely or never used calculators. But in the US
and 10 of 11 nations with scores below the international average, many used
calculators every day.
"Down the highway from Okemos is the city of Portland, where Portland Middle
School is a pilot for [newnew math program] Connected Math. A sixthgrader recalls that at his
elementary school kids used calculators in every grade. 'I can't even do division
without a calculator,' he says. 'Last year we did this weird type of division, I
think they called it 'long' ... I didn't really get it. The teacher told us
'don't worry, you're doing the work. The calculator is just showing you the
answer.''"

"a dramatic loss of insight"

Dangers of the Paradigm Shift by Jose MiroJulia: This very interesting paper on math education
considers the positive role of calculators, but worries about a profound "loss of knowledge"
from their misuse. Among the author's concerns are students thinking of
all operations as being of equivalent utility (repeated addition on a calculator
requiires little more effort than multiplication), a dramatic loss of insight into the subtleties
of a mathematical problem, and the blind faith in the "sorcery" of the calculator.
The author surmises that calculators inhibit rather than encourage critical thinking.
 A group of mathematicians at Northwestern University offers this lovely item for sale:

Mathematicians agree 93to0:
Basic math, not calculators

Math professors 93to0 in supporting basic math:
Stephen Wilson, professor of mathematics at Johns Hopkins University, asked a number of mathematicians
their thoughts on the following statement:
Statement:
"In order to succeed at freshmen mathematics
at my college/university, it is important to have
knowledge of and facility with basic arithmetic
algorithms, e.g. multiplication, division, fractions,
decimals, and algebra, (without having to rely on
a calculator)."

The response was overwhelming and unanimous: 93 mathematicians agreed with the statement, and NONE disagreed!
Prof. Wilson comments on these results,
"This is particularly remarkable because if you ask this same
bunch what mathematics is, then no two of them will agree.
There are at least a couple of dozen who would normally
disagree just to be disagreeable."
Click the link to see their affiliations and many of their extended comments, but here are some of the highlights:
 Professor, Wayne State University:
"That it is even slightly in doubt is strong evidence of very distorted
curriculum decisions. I do not know even one universitylevel teacher of
mathematics who would disagree with it. I would be truly astonished to
meet a person who disagrees."
 Professor, University of Calgary:
"Last winter at this time I was visiting Chennai (Madras), and one of
my talks was at a Jr H S (called a Secondary School there). I was amazed
to see the facility these young students had with arithmetic  of course
they never use calculators at all in school."
 Math Ph.D., Union College:
"It is essential to have knowledge of and facility with basic arithmetic
algorithms, e.g. multiplication, division, fractions, decimals, and
algebra, (without having to rely on a calculator). Students without this
ability typically do not make it successfully through their introductory
calculus courses, and are often forced to repeat courses or to drop out of
engineering/science programs."
 Professor, University of Illinois:
"Here is a new twist. My daughter got a terrible score
on her SAT II or whatever, because her calculator's
battery was dead, so she borrowed her brother's,
but it was set to give exact answers, and she didn't
know how to convert them to decimals, and she
was too stressed to figure it out, so she couldn't
do multiplechoice answers that required knowing
whether pi or e (say) was closer to 3."
 Professor, Northern Illinois University:
"No doubt you are aware that the US educational system is releasing
increasing numbers of students who fail to meet even the minimal
standards imposed by state boards of review. ... It's a bad situation
and I'm glad you're standing up for a minimal competence."
 Associate Dean and Professor of math, Queen's University [Ontario]:
"It makes an enormous difference to have students who can add fractions, for
example. This last skill is a key indicator of success  if you cannot add
fractions, you do not belong in university  we do not have the time nor the
resources to teach you that which you should already know on being accepted.
If you have graduated from high school without such skills, your high school
has cheated you. In my view, the teaching of basic arithmetic skills is not an option for
schools, but rather an important part of their mandate. I'd be very unhappy
to send my children to a school that thought otherwise."
 Assistant professor math, Penn State:
"Having taught 20 years in public school mathematics and now 16 years at
Penn State Altoona in the Mathematics Dept. I heartily agree that public
school students must learn the basic arithmetic algorithms to be successful
in college mathematics courses. Calculators are a good thing and are being
used extensively in my engineering math classes, but successful students
know the basics without a calculator."
 Professor, Fordham University:
"I more than agree with your statement about the need for
children to learn arithmetic; and the necessity of being
able to do simple arithmetic without a calculator. ...
I see many students who, when confronted with an expression
like (64)^(2/3) will hit their calculators to find out the
value; but, because they have been raised with the calculator,
have no idea what the expression means; how it has been defined;
what are the algebraic properties of exponents."
 Ph.D., University of Goettingen [Germany]:
"What a question: the answer is of course 'yes, obviously'!!!!"
 Professor, Univesite Louis Pasteur [Strasbourg, France]:
"I completely agree with your statement ... It is sad that such things which ought to be completely obvious
re controversial!"
(Click here for the full list of math professionals and their comments.):
Equity and Calculators

Calculators May Be the Wrong Answer As a 'Digital Divide' Widens in Schools
by Daniel Golden, Wall Street Journal, Friday, December 15, 2000.
"Teachers like Mr. Martin favor calculators as motivational tools. ...
But more calculator use in innercity schools generally hasn't added
up to higher test scores. The majority of experts on
elementaryschool learning maintain that, for students who lack basic
number proficiency, calculators may provide only the illusion of
progress. 'Kids get to use calculators as a substitute for practice,
and they never really understand arithmetic,' says Sandra Stotsky,
deputy education commissioner in Massachusetts, a state that has
taken a backtobasics approach."

Do Other Schools Have an Unfair Advantage on SATs and AP Exams?
by Sarah Brown, CoEditorinChief, Cougar Crier, John F. Kennedy High School, Bellmore, New York, December 2002.
This is a remarkable article, not just for the depth of its research and quality of its writing,
but also for the fact that it was written by a high school student
for her school newspaper. Excerpts:
"Most Kennedy students are unaware that upperlevel math classes in
Port Washington, Jericho, Great Neck, and other high schools across
Long Island and the country are using the Texas Instruments89
calculator, a far more advanced calculator than that which is used at
Kennedy. ...
Many students, parents, and teachers complain about schools' usage of the TI89,
claiming that the calculator inhibits learning.
'The kids don't have to know as much,' admitted
[Great Neck North High School's Mathematics Chairperson Helen] Kramer."

Pricey Calculators Give Some Edge In Math Tests
by Nedra Rhone, Newsday, November 23, 2003.
"Staples office supply store was a madhouse during the late summer
backtoschool shopping crush. ... Mothers were screaming, 'I didn't know the calculator was
$90!'
The calculator, a graphing handheld listed among the necessary class
materials for Deer Park High School and others on Long Island, earned
a place in the great Regents Math A debate. While the issues of
content and grading dominated educators' discussions after an
estimated onethird of students in the state failed the exam last
June, the use of the pricey calculator that sent some parents into a
frenzy raised questions of equity and stirred ageold debates about
the use of technology in math instruction.
Students who are able to afford graphing calculators, or who live in
districts that provide the tools, have a distinct advantage over
other students if permitted to use the calculator on a Regents exam, reported an independent panel of educators last month."

Math Teachers Debate The Use Of HighPowered Calculators
by Lisa Black, Chicago Tribune, November 3, 2003.
"A handheld calculator that can solve brainnumbing
algebra equations within seconds has high school math teachers divided
over whether it will make algebra more accessible or rob students of
basic skills."
 KnockKnock: Its the C.E.O.,
New York Times, April 12, 2009,
interview with Terry J. Lundgren, chief executive of Macy's. Excerpt:
Q. Anything you would like business schools to teach more? Less?
A. In our business, there's not enough emphasis on math. Coming out of
college, we really like to have kids who like math, study math and get it.
And so I'd like to make sure that there is an emphasis on math. I think
there is a strong emphasis on marketing already, and we want that and we
need that. But to me, the math piece is weak in most business school
educations, and I'd like to have more emphasis on that.
Q. But somebody might say, "That's what calculators are for."
A. And that's exactly the problem. ... I think there's logic that has to go into
this. And I don't think you should actually have to have a calculator for
every decision that you make that has numbers attached to it. Some of that
should just come to you quickly, and you should be able to quickly move to
your instincts about that being a good or not good decision."
Comments on Calculators
Algebra
"GuessandCheck" Is Not Algebra!
 Parents love it when they see their kids assigned "prealgebra" problems like this:
Two numbers add up to 15, and the difference between them is 3. What are they?
This is a simple problem  with algebra. But what a mess it turns out to be when
educrats use this problem to introduce "Guess and Check" to grade schoolers as a "problem solving strategy".

PseudoEducation Marches On by Domenico Rosa, December 4, 2004.
"Guessandcheck appears to be the 'strategy' of choice
that is being taught for solving simple problems. ...
These 'strategies' are becoming more and more widespread under the guise
of 'solving algebra problems' and 'algebra for all.' In my opinion, far
from teaching any meaningful concepts, these mechanical calculations are
doing little more than enhancing the pseudoeducation of American
students.
This type of pseudoeducation is being promoted  at conferences,
workshops, minicourses, and training sessions  by assorted 'experts' who
promise to boost scores on assorted 'mastery tests' and other
standardized tests. These promotions are being adopted mindlessly by
administrators and teachers, whose bonuses and other financial rewards
are based on the results of these tests.
As long as this rampant pseudoeducation continues to be promoted, the
situation in the U.S. will only worsen."
Basic Math is a Prerequisite for Algebra

California's Algebra Crisis
by Paul Clopton and Bill Evers, October 6, 2003.
"The strongest predictor of failure to learn algebra is not race or income;
it is a lack of adequate academic preparation.
The problem begins before students get to their first algebra class.
Many school districts have watered down the content of prealgebra courses,
removing important but difficult material. The districts want more
students to pass math classes, and they want to guarantee high
pass rates by making the classes easy. But classes without content
set students up for later failure in algebra. ...
Admirably, California embraces learning algebra by the end of eighth grade
as a longterm goal. But strengthening academics from kindergarten on is
necessary before this goal can fully be met.
Algebra placement rates ought to depend on student readiness."
 The Dumbing Down of Algebra
by C.F. Navarro, Ph.D. Excerpts:
"At the George Washington Middle school where I taught eightgrade
math in 1998, only a few of my math students were at grade level. The
rest were at a fourthgrade level, or lower. ... Most had not yet learned
their multiplication tables and were still counting with their
fingers. By the end of the year some had progressed to about a
fifthgrade level, a substantial improvement, but far short of the
comprehension and skills required for algebra. Nonetheless, all were
required to register for algebra the following year.
"More troublesome still was my algebra class. ...
with few exceptions, they didn't know how to work with fractions,
decimals or integers. They lacked the power of concentration to set
up and solve multiplestep problems. They were incapable of
manipulating symbols and reasoning in abstract terms. Like most of my
general math students, some had not yet learned their multiplication
tables and were still counting with their fingers. All had been
issued graphing calculators (a terrible mistake) and led to believe
that algebra consisted simply of pushing buttons and getting the
right answers.
"Given another year or two to mature and learn their basic math, most
would have mastered algebra and gone on to higher mathematics without much trouble.
But as it turned out, all they got from their premature exposure to algebra
was a lot of stress. Some, I suspect, will hate math as long as they live.
"The education establishment, however, is not wont to give up a bad idea.
If it cannot bring the kids up to algebra, then it will bring algebra
down to the kids. ...
But no matter how much the subject is fragmented and, in the process,
dumbed down; no mattter how many howtoteachalgebra workshops high school
teachers are forced to take, students unprepared for the subject
are not going to learn it. ...
The early algebra and algebraforall program in our public schools
looks great on paper. It gives the impression that our local kids
have finally caught up with their counterparts in Japan and Norway.
But in truth, they are just as far behind as ever."

What Does It Take To Learn Algebra? First You Have To Master The Fundamentals
by Karin Klein, Los Angeles Times, February 4, 2006.
"Things looked pretty hopeless to both of us those first couple of
sessions, as Johnny stumbled through algebra problems while I tried
to figure out exactly what he didn't understand. Then, as we took it
down to each step of each little calculation, the trouble became
clear: Johnny somehow had reached ninth grade without learning the
multiplication tables.
Because he was shaky on those, his long multiplication was error
prone and his long division a mess. As Johnny tried to work algebraic
equations, his arithmetic kept bringing up weird results. He'd figure
he was on the wrong track and make up an answer.
This discovery should have made us feel worse. How could we possibly
make up for a dearth of thirdgrade skills and cover algebra too?
But at least we knew where to start. ...
in all these decades, the same school structure that failed Johnny
goes on, dragging kids through the grades even if they don't master
the material from the year before. This especially makes no sense for
math, which is almost entirely sequential."

Things Don't Add Up In B.C. Math Classes
by Bill Hook and Karin Litzcke, Vancouver Sun, Issues & Ideas, March 04, 2005.
"... because elementary math fails to provide a solid foundation, many
basically capable students simply give up when faced with the shock
of high school algebra, which would be the doorway to advanced
technical training at all levels. ... [T]eachers cannot
make up Grades 1 to 7 while teaching Grade 8."
Is Algebra Needed?
 Our own Dave Ziffer defends the teaching of algebra, in this open letter to
Education Week magazine. Excerpt:
"Mr. Bracey contends that algebra is an essentially useless skill,
malevolently imposed upon our students for the purpose of sorting out which
children will attend college. But wait a minute, don't all of us in the
scientific and mathematical communities emphatically claim that algebra is
the very foundation of the work that we do every single day? Not so,
according to Mr. Bracey. Apparently he thinks that people can build bridges,
produce electricity, design cars, and fly airplanes without any use of
algebra at all."

Algebra Benefits All Students, Study Finds
by Kathleen Kennedy Manzo, Education Week, November 15, 2000.

Algebra and Its Enemies
by Kenneth Silber, May 8, 2006
Learning Algebra

The Effects of Cumulative Practice on Mathematics Problem Solving
by Kristin H. Mayfield And Philip N. Chase,
Journal of Applied Behavior Analysis, Summer 2002.
"According to the latest international study, the average
score of U.S. students was below the international average, and the top 10% of U.S.
students performed at the level of the average
student in Singapore, the world leader. In addition, recent tests administered by the U.S. National Assessment of
Educational Progress revealed that 70% of
fourth graders could not do arithmetic with
whole numbers and solve problems that required one manipulation. Moreover, 79% of
eighth graders and 40% of 12th graders
could not compute with decimals, fractions,
and percentages, could not recognize geometric figures, and could not solve simple
equations; and 93% of 12th graders failed to
perform basic algebra manipulations and solve
problems that required multiple manipulations.
These statistics reveal students' deficits in
the fundamental skills of mathematics as
well as mathematical reasoning and problem solving."
Success in Low Income Schools
 RECOMMENDED:
High Achievement in Mathematics: Lessons from Three Los Angeles Elementary Schools
by David Klein,
Commissioned by the Brookings Institution, August 2000.
"What can elementary schools do to promote high achievement in mathematics for
their students? While much knowledge has been gained in recent years about the
teaching of reading, relatively little is known about what constitutes effective
mathematics programs. This paper describes characteristics and policies of three low income schools with
unusually successful mathematics programs in the Los Angeles area."
Quotes on Math Education
From our extensive page on education quotations:
"I recommend you to question all your beliefs,
except that two and two make four."
 Voltaire (L'homme aux quarante écus)
"Strange as it sounds, the power of mathematics rests on its
evasion of all unnecessary thought and on its wonderful saving
of mental operations."
 Ernst Mach, physicist
"A mastered algorithm in the hands of a student is an incomparable tool
laying bare the conceptual structure of the mathematical problems that
the algorithm solves. With such tools, and with the guidance of good
teachers in their use, a student can grasp and integrate in twelve
years a body of mathematics that it has taken hundreds of geniuses
thousands of years to devise."
 David Ross, Ph.D., a mathematician at Kodak Research Labs
"Piaget's constructivism and Bourbaki's austere rigor have left their marks
on our schools. Will such trenchant educational theories ever give way
to more serene and better optimized teaching methods, based on a genuine
understanding of how the human mind does mathematics?"
 Stanislas Dehaene, "The Number Sense: How The Mind Creates Mathematics"
"Discovery lessons, students writing to learn mathematics, the teaching of
socalled general problemsolving concepts, field trips, math lab lessons,
alternative assessments, collaborative partner tests, student presentations, and
openended problems should all be used sparingly. I use some of them, but they
have limited value. Penciland paper analytic solutions are the heart of
mathematics education."
 Michael Stueben,
Twenty Years Before the Blackboard
"These thoughts did not come in any verbal formulation. I rarely
think in words at all. A thought comes, and I may try to express
it in words afterward."
 Albert Einstein
Dr. Einstein would have done lousy on "performance assessment" tests
with that attitude!  Editor
"It's been my experience that students in secondary math education
... are generally among the worst students in my class.
The background in math of prospective elementary school teachers
is even worse, in many cases nonexistent."
 John Allen Paulos, Innumeracy
"The mathematics background that elementary school teachers typically
receive is atrocious  little or none"
 Paul Sally Jr., professor in mathematics, University of Chicago
"Nothing flies more in the face of the last 20 years of research than the
assertion that practice is bad. All evidence, from the laboratory and from
extensive case studies of professionals, indicates that real competence only
comes with extensive practice. ... In denying the critical role of practice one
is denying children the very thing they need to achieve real competence."
 John R. Anderson, Lynne M. Reder and Herbert A. Simon, Carnegie Mellon University,
in Applications and Misapplications of Cognitive Psychology to Mathematics Education
"...varied and repeated practice leading to rapid recall and automaticity
is necessary to higherorder problemsolving skills in both mathematics
and the sciences. ... lack of automaticity places limits on the
mind's channel capacity for higherorder problemsolving skills. ...
only intelligently directed and repeated practice, leading to fast,
automatic recall of math facts, and facility in computation and
algebraic manipulation can one lead to effective realworld problem solving.
... [These conclusions are based on] reliable facts, figures,
and documentation ... not just from isolated lab experiments, but
also from largescale classroom results."
 E. D. Hirsch
"Computational algorithms, the manipulation of expressions, and paperandpencil drill must no longer dominate school mathematics."
 National Council of Teachers of Mathematics (NCTM), Professional Standards for Teaching Mathematics, 1991
"The NCTM denigrates the idea of practice, which is thoughtful, considered repetition,
and confuses it with drill, which is blind, mindless repetition."
John Saxon
"If you have one bucket that contains two gallons and another bucket
that contains seven gallons, how many buckets do you have?"
 math problem in the movie Idiocracy
"Numbers rule all things."
 Pythagoras
"Never underestimate the joy people derive from hearing something they already know."
 Enrico Fermi (19011954)
"In the new approach, as you know, the important thing is to understand
what you're doing rather than to get the right answer."
 Tom Lehrer, New Math
"So you've got thirteen ...
And you take away seven,
And that leaves five...
well, six actually, but the idea is the important thing."
 Tom Lehrer, New Math
"Presumably no one would argue that the conservative view on the
sum of 14 and 27 differs from the liberal view."
 Carl Sagan, The DemonHaunted World, page 257
"Mathematical discoveries, small or great are never born of spontaneous
generation. They always presuppose a soil seeded with preliminary knowledge and
well prepared by labour, both conscious and subconscious."
 Jules Henri Poincaré, mathematician (18541912)
"To those who do not know mathematics it is difficult to get across a real feeling
as to the beauty, the deepest beauty, of nature... If you want to learn about
nature, to appreciate nature, it is necessary to understand the language that she
speaks in."
 Richard Feynman
"Mathematics is the language in which God has written the universe."
 Galileo Galilei
"This notion that one has to 'interest' students in mathematics in
order to make them do it has gone much too far, to the point where
real mathematics in many cases has just disappeared entirely from the
courses. They're just a discussion of what mathematics does and
beautiful pictures and imprecise ideas."
 Paul Sally Jr., professor in mathematics, University of Chicago
"Belief is no substitute for arithmetic."
 Henry Spencer
"Newsrooms are full of English majors who acknowledge that they are not good at
math, but still rush to make confident pronouncements about a globalwarming
'crisis' and the coming of bird flu."
 John Stossel
"My whole experience in math the last few years has been a struggle
against the program. Whatever I've achieved, I've achieved in spite
of it. Kids do not do better learning math themselves. There's a
reason we go to school, which is that there's someone smarter than us
with something to teach us."
 High school student Jim Munch, relating his battle against his school's constructivist math program
"It is India that gave us the ingenious method of expressing all numbers by means of ten symbols,
each symbol receiving a value of position as well as an absolute value;
a profound and important idea which appears so simple to us now that we ignore its true merit.
But its very simplicity and the great ease which it has lent to computations put our arithmetic
in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement
the more when we remember that it escaped the genius of Archimedes and Apollonius,
two of the greatest men produced by antiquity."
 PierreSimon de Laplace (17491827)
"Well, the math stuff I was fine with up until 7th grade. But Malia is now a freshman in High School and IÕm pretty lost. ItÕs tough."
 Barack Obama, The Tonight Show, 2012
More Excellent Resources on Math

Weapons of Math Destruction: Cartoons about fuzzy math!
 Resources on
mathematics education, assembled by Dr. Martin Kozloff

Math Reform Issues:
an outstanding collection of articles and reports on the math controversy,
presented by the Oregon Education Coalition.

Kids Do Count: Seeking Excellence in Math Education is an extensive (and beautifully designed) website
organized by a group in Utah protesting the use of fuzzy math there. They provide excellent
commentary and personal histories on problems, and a very rich set of links for additional information.
 How well are American students prepared in math? Decide for yourself: take a look
at these examples of math problems from Japan's
University Entrance Center Examination (UECE) (PDF doc).

The Math Forum at Drexel University: An extensive and fascinating collection of articles and links
on math teaching issues, assessment, computers and calculators, research, education reform,
societal issues, public policy, and public understanding of math.
 From our page on books on education, see this section on
math.
