The Fuzzy Math Mindset Behind
Phil Mickelson ExxonMobil Math
Each summer, elementary
school teachers attend Phil Mickelson ExxonMobil
Academies where they are to learn "best practices" for
"encouraging children's interests in math and science." Math Solutions
Professional Development, founded
by Marilyn Burns, will supply the "math"
staff. What's wrong
with this picture? Burns makes elementary math "interesting" by
discarding standard paper-and-pencil arithmetic and replacing it with
calculators and "math appreciation" activities. Bright kids are
bored by the "hands-on" busywork, and teachers attending Mickelson
Academies won't find
anything new. They're quite familiar with the 100-year-old
philosophy promoted by Marilyn Burns. Currently known as
"constructivism," it's the progressive "discovery-learning" methodology
that's responsible for the the current dumbed-down state of American
Fuzzy math won't be adequate for children who will eventually seek
employment as carpenters, plumbers, chefs, or in many other occupations
that don't require a college education. But the fuzzy math
deception is particularly cruel for children who will eventually attend
college, because the K-6 math that was taught 50 years ago is the basic
math that college-bound
children still need to
80% of K-6 math education should still be devoted to the mastery of
arithmetic. Without such
mastery, a child has no hope for later mastery of non-fuzzy
algebra. Fuzzy "reform algebra" won't do. It's synonymous
with applications of a graphing calculator. Such technical skills
won't help much with college math. Mastery of traditional algebra is still
the gateway to
learn how strongly university
mathematicians feel about the importance of standard arithmetic and
traditional algebra, click on the survey of math professors conducted by Steve
Wilson, Professor of Mathematics at Johns Hopkins.
The following clickable
internal document links also serve as an outline for
The short answer is through the
traditional methods of teacher-guided knowledge transmission.
Math is the first knowledge domain where American children build a
remembered knowledge base of domain-specific content, with each child
gradually coming to understand hundreds of specific ideas that have
been developed and organized by countless contributors over thousands
of years. With teachers who know math and sound methods of
knowledge transmission, the student is led, step-by-step, to remember more and more
specific math facts and skills, continually moving higher and higher up
the vertically-structured knowledge domain that comprises traditional
K-6 math. This first disciplined knowledge-building experience is
a key enabler, developing the memorizing and organizing skills of the
mind, and thereby helping to prepare the child for more advanced
math learning, and also helping to prepare the child to eventually
remembered knowledge bases relative to other knowledge domains in the
professions, business, or personal life.
Under the knowledge transmission model, children read textbooks and
listen to teachers. Teachers lead the whole class, asking
questions and presenting problems which have been carefully chosen to
lead students to understand teacher-targeted knowledge. Content
is taught directly in a step-by-step manner, gradually moving from the
simple to the complex. Feedback and correction are
immediate. Standard content is being transmitted, so
standardized tests are used. [Constructivists
regularly suggest that
better than "standard." This may be true for arts and crafts, but
standards and conventions are essential for mutual understanding and
communication in business, science, and professional life.]
Memorization and practice are
regularly used knowledge retention techniques. Beyond basic
knowledge retention, these techniques help the process of integrating
knowledge fragments to form concepts. They also promote a gradual
transition from conscious thought to automatic use and thereby free the
conscious mind to focus on higher-level learning. Knowledge
retention is necessary for mathematical understanding and
math knowledge is pulled together, connected in the mind, and applied
to solve the current problem. External knowledge "look ups" may
involved, but remembered knowledge provides the necessary orienting
framework of background information. Remembered background
to the recognition of what should be looked up, and it leads to the
recognition of how the looked-up knowledge can used.
is most difficult when the mind is initially empty of content.
Understanding becomes easier and increasingly faster as we remember
(store in the
brain) more and more specific math facts and skills.
Should Children Learn in Elementary Math?
- Automatic recall of
- This is the key necessary condition for later mastery of the
standard algorithms for multidigit computation. For example, the
second grader should instantly know that 7 + 8 = 15, and the fourth
grader should instantly know that 7 x 8 = 56. Students who don't
automatically know single digit number facts will get bogged down
when they encounter the multidigit computational algorithms. By
ingenious design, these algorithms reduce any mulitidigt computation to
of single digit facts.
- Example: The standard algorithm
for multidigit multiplication reduces 35 x 78 to 4 single digit
multiplication facts [5 x 8, 5 x 7, 3 x 8, 3 x 7].
algorithm for multidigit addition is then used to add the two partial
products, 390 and 2,340. This addition is carried out using
single digit addition facts, 4 + 9 and 3 + 3. As is
frequently the case in
mathematics, mastery at higher levels depends on mastery at one or more
- Instant recall of single-digit number facts is also a
necessary condition for later mastery of operations with
fractions. The ability to add, subtract, multiply, and divide
fractions also depends on prior mastery of multidigit addition,
subtraction, and multiplication. Once again, building
knowledge at a given level depends on knowledge built at one or
more lower levels.
- Click on FlashMaster
to learn about this effective electronic flashcard device.
of the Standard Computational Algorithms for Numbers in All Forms:
By the end of elementary school,
students should have mastered the standard methods for adding,
subtracting, multiplying, and dividing whole numbers, fractions, and
decimals. They should understand why the algorithms work, and
they should be able to carry out algorithm steps automatically, without
Such mastery frees the mind to focus on higher level tasks.
- Word Problems:
By the end
of elementary school, students should be able to solve
traditional 2 and 3 step word problems.
- Example: Sarah spent 2/3 of
her money on a book and a CD. The CD cost 3 times as much as the
book. If the CD cost $24, how much money did she have left?
By the end
of elementary school, students should know how to find the perimeter
and area of basic 2-dimensional geometric shapes, and they should know
how to find the surface area and volume for cubes and cuboids.
- Language: By the
elementary school, students should know how to communicate elementary
math information by the using the precise language and symbols of
standard arithmetic and basic geometry.
If a student must rely on a calculator for arithmetic, then later
of genuine algebra is ruled out.
- For more specific
as to what a child should know at a specific grade level, see the placement tests
For the best available program for elementary math, see Primary
Mathematics, U.S. Edition.
Marilyn Burns View of How Children
Should Learn "Math"
Marilyn Burns has
books dealing with elementary math education. In Part I of About
Teaching Mathematics: A K–8 Resource,
Second Edition she presents her constructivist view of how children
should learn math. The fundamental assumption is that children
learn best through "hands-on" attempts to solve "real world
problems." Marilyn Burns writes: "continuous interaction between
mind and concrete experiences with mathematics in the real world is
necessary." Preaching constructivist gospel, Burns wants
teachers to be guides on the side and avoid direct instruction.
Children are to work in small groups,
regularly seeking help from others in the group, and only asking the
teacher for help, if all members of the group have the same
question. She writes
"rather than directing a lesson, the
teacher needs to provide time for students to grapple with problems,
search for strategies and solutions on their own, and learn to evaluate
their own results." And "teachers need to urge students to find
ways to verify their solutions for themselves, rather than rely on the
teacher or the answer book." According to Burns, "in real life
it's up to the problem solver to decide when a solution is 'right' or
'best'." She cautions that such "natural learning" takes
significant time, and "it's essential that teachers provide the time
that's needed for children to work on their own and that teachers not
slip into teaching-by-telling for the sake of efficiency." How to
evaluate what children have learned? Burns writes "a teacher can
assess from listening in whole class discussions, observing during
small group work, and reading students' work."
Constructivists call this "authentic assessment."
("teaching-by-telling") , step-by-step
feedback and correction, and knowledge retention techniques are all
strongly discouraged. It's expected that all
learning will occur naturally as a byproduct of games, investigations,
and other small group "problem-solving activities." How are
problems to be solved?
On page 15 Marilyn Burns offers
the constructivist "Problem-Solving Strategies." Children are to
a pattern, construct a table, make an organized list, act it out, draw
a picture, use objects, guess and check, work backward, write an
equation, solve a simpler (or similar) problem, or make a
Marilyn Burns Rejects Standard Paper-and-Pencil
Burns is vague about specific math learning objectives. But
she's clear that children shouldn't learn standard arithmetic.
Here are quotes
Arithmetic, Part III of in
Teaching Mathematics: A K–8 Resource.
- "Facility with standard
arithmetic is no longer the measure of arithmetic understanding
and competence." Page 139
- "Because of the present
availability of calculators, having
children spend more than six years of their schooling mastering
paper-and-pencil arithmetic is as absurd as teaching them to ride and
care for a horse in case the family car breaks down."
- "There is no way for all
students to do arithmetic calculations in the same way any more than it
is essential for all children to develop identical handwriting or
writing styles." Page 153
- "the emphasis of arithmetic
instruction should be on having students invent their own ways to
compute, rather than learning and practicing procedures introduced by
teacher or textbook." Page 154
- "The change from teaching
time-honored algorithms to having children invent their own methods
requires a major shift for most teachers. It requires, foremost,
that teachers value and trust children's ability and inventiveness in
making sense of numerical situations, rather than on their diligence in
following procedures." Page 156
- "In all activities, the
emphases are on having children invent their own methods for adding and
subtracting . . . the standard algorithms are not taught." Page
- "Also, rather than teaching the
standard computational algorithm for multiplying, the activities give
students the challenge of creating their own procedures for
computing." Page 194
- "A great deal of emphasis
traditionally has been put on paper-and-pencil algorithms for addition,
subtraction, multiplication, and division of fractions. Too much
often on 'how to do the problem' rather than on 'what makes
sense'. The following suggestions offer ways to have
students calculate mentally with fractions.
The emphasis shifts from pencil-and-paper computation with the goal of
arriving at exact answers to mental calculations with the goal of
at estimates and being able to explain why they're
reasonable." Page 232 [Bold emphasis added].
On page 241 we have a statement that is almost identical to quote
#8, the quote just given for fractions. The difference? Two
occurrences of "decimals"
replace the two occurrences of "fractions."
Mastery of exact computations with
fractions is the number one predictor of later success in algebra.
By discarding standard paper-and-pencil
arithmetic, Marilyn Burns eliminates
about 80% of traditional elementary math education
content. This conveniently
leaves plenty of time for discovery-learning
"problem-solving activities." These time-consuming
include explorations of advanced topics such as algebra, probability
and statistics. But this is an obvious sham, because math
domain. Algebra, probability and statistics are ruled out, if the
hasn't first mastered standard paper-and-pencil arithmetic. We're
talking about genuine algebra, probability and statistics, not the
constructivist fuzzy math versions.
Marilyn Burns promotes "problem
solving as the focus of math teaching." In her book, About
Teaching Mathematics: A K–8 Resource, she offers many examples of
problem is given on page 71. The Marilyn Burns solution is found
on pages 266 - 267. Here's a brief statement of the
problem: There are 3 cards in a paper bag: one with an X
on both sides, one with an O on both sides, and one with an X on one
side and an O on the other side. If you draw one of these cards
random and look at only one side, how would you predict what is on the
other side? Marilyn
Burns offers her
solution on page 266. She writes "I found choosing a
strategy for this problem perplexing." She then explained that
she first reasoned that there are a total of 3 Xs and 3 Os, so
if she sees an X, then 2 Xs and 3 Os are left. Since there
are more Os left, she will predict the other side will be an
O. More generally, her strategy is to predict the
opposite of what she sees. Marilyn Burns then tested
this strategy by playing 30 games [30 experiments of randomly
drawing 1 of the 3 cards]. She found that her prediction
correct less than half the time, so she played another 30
games. Same result. Not satisfied, Marilyn
Burns was then advised by a colleague that she should predict what she
sees, not the opposite. The new logic: "there's a 1/3 chance that
you'll draw a card with an X on one side and an O on the other side,
and a 2/3 chance that you'll draw a card with the same mark on both
sides. A probability of 2/3 is twice as likely as a probability
of 1/3, so predict the mark you see."
Marilyn Burns tested this
new strategy by again playing two rounds of 30 games. This
time she was happy with the results and proclaimed "case closed."
- These activities aren't accompanied by a statement of appropriate
grade level, prerequisite math knowledge, or math learning
- Although never
explicitly stated, prerequisite math knowledge is often
needed for the Marilyn Burns "problem-solving activities." How
did the child acquire the prerequisite math knowledge needed to solve
the current problem? Constructivists would have you believe
that the child somehow picked up this knowledge while carrying out
earlier "problem-solving activities."
- The Burns problem-solving activities" often consume considerable
time, because they
require extensive busywork and regular use of a wide
variety of "hands-on" concrete objects.
- Example: "Find
all the different ways to arrange four
toothpicks by the
following two rules: 1. Each toothpick must touch the end of at least
one other toothpick. 2. Toothpicks must be placed either end to end or
to make square corners." Page 92
What's the point? This question needs to be
asked for every Marilyn Burns "problem-solving
- Marilyn Burns provides few samples of student
solutions. She regularly states that there are many possible
solutions. But she does offer her solutions to some problems
IV [pages 255 - 308] of About
Teaching Mathematics: A K–8 Resource. Here are four
examples of her solutions:
Comment 1: For a mathematically correct
discussion of this problem, see Conditional
Probability. For a more informal discussion and links to
variations , see the Three Cards
A Tangram Problem
why it's not possible
to form a square using exactly 6 of the 7 Tangram pieces.
This problem is given on page 83. The Marilyn Burn solution
is discussed on pages 274 - 277. There she discusses how she
formed a panel of 4 Math
instructors. They were to help her solve
She writes: "we observed four people rummage for ways to approach the
problem. They cut Tangram pieces; they moved pieces
about; they exchanged ideas; at times, one person would
retreat into private thoughts and then reemerge to share
discoveries." Marilyn Burns later says "I thought about this
problem for a long time - years - before finally making sense of
it for myself." Years!
Burns appears to believe
that 60 trials proves the correctness of her solution, and her "math"
staff appears to
agree. But an experiment with a small number of
trials [such as 60] may not result in a experimental probability that
is an excellent approximation for the theoretical
Marilyn Burns and her staff spent considerable time trying to "solve"
this problem with a
"hands-on" concrete materials approach. After 3 pages of
discussion, she quietly discards hands-on methods and gives a
traditional solution. The idea is to subtract the area of
the excluded 7th piece from total area of all 7 pieces. This
the area of the potential 6-piece square. Then take the
square root of the 6-piece area to
get the length of the 4 sides of the potential 6-piece square.
Then show that this length
can't be formed using sides of the 6 pieces. Due to congruent
pieces, there are 4 (not 7) cases to consider.
Comment 2: The
successful method requires prerequisite knowledge about area and the
length of sides for squares, triangles, and parallelograms.
This knowledge can't be discovered while the student attempts to
solve this problem.
Trace around your left foot on centimeter squared paper. Find the area
of your foot. Cut a piece of string so that the length is equal
to the perimeter of your left foot. Tape the string in the form
of a square on centimeter squared paper. How does the area
of your foot compare to the area of the square? This problem is
given on page 54. Marilyn Burns discusses it on page 256.
She writes: "I was mathematically flabbergasted the first time I
encountered this problem," because "I believed that two shapes with the
same-length perimeter should have the same area." After
several hands-on investigations, her "understanding
shifted." But she recommends: "don't take my word for it.
There's no substitute for firsthand experience, so try some
investigations for yourself."
Comment 1: If you know
about the perimeter and area of rectangles, you may quickly
find a counterexample to the "same
perimeter implies same area" conjecture.
For example, consider any rectangle with width = W and , length = 50 -
W. The perimeter is always 100, but the area can be as large as
625, if W = 25, and the area can be as close to 0 as desired, if W is
sufficiently close to either 0 or 50.
Comment 2: Marilyn Burns
"investigations," so she doesn't appear to know that one
counterexample suffices to prove that the conjecture is
Box Measuring Given a
20-by-20 centimeter piece of centimeter-squared paper, you can cut a
square from each corner and fold up to form an open-top
box. How many different size boxes can you make using this
method? Which of these boxes holds the most? The
problem is given on page 55. Marilyn Burns discusses it on page
page 257. She presents a 9 row table, giving the dimensions
and volume for the 9 whole number cases. The largest volume is
given as 588. This is found on the 3rd row [case that the side of
the cutout corner square is 3 cm]. The next largest volume is
given as 576. This is found on the 4th row [case that the side of
the cutout corner is 4 cm]. Marilyn Burns conjectures that
the largest volume is somewhere between these two cases. She
3.5 cm and gets a volume of 591.5. Looks
promising! But she then ends the discussion by suggesting
"you may want to investigate what size square to cut from each corner
get the box of maximum volume." She quit!
was the only choice. Marilyn Burns isn't going to solve this
problem with her
"guess and check" approach. There are infinitely many
possibilities for "the different size boxes you can make," and kids
aren't going to easily discover "which of these boxes holds the
most." This isn't an appropriate problem for K-6 math or even
K-10 math, but it's a simple problem in differential
calculus. The volume V = H (20 - 2H)2. The
derivative V' = 400 - 160H + 12H2 = 4 (10 - 3H) (10 -
the maximum occurs when H = 3 1/3.
Notice the prerequisite math knowledge and the time-consuming busywork
needed to produce Marilyn Burns' 9-row table. Also, notice that
if the "find the maximum volume" challenge was taken seriously, kids
could spend endless hours and never know for sure that they had found
the correct answer.
from the Marilyn Burns
If discovery-learning is taken to its
logical extreme, fewer teachers would be needed. Marilyn
Burns can't endorse that. And she's been around long enough
to know that math isn't discoverable. It's not natural.
It's an invented knowledge domain The no "teaching-by-telling"
fuzz is convenient for incompetent teachers, but awkward for someone
who is trying to make a living as a teacher of teachers.
hoping that constructivist zealots won't notice the contradiction,
Burns promotes whole class
model lessons. Here
are four 5th
grade examples from her website.
- The first problem is to separate a 5 3/4 pound bag of candy
pound bags. How many 1/2 pound bags
will this yield? Three solutions are given. This is a very
easy mental math question. One student's solution correctly uses
fraction division to find the answer of 11 of the small size bags,
leaving an extra 1/4 pound of candy. The other two
students use constructivist problem-solving
strategies. These methods involve drawing pictures showing 5
boxes divided into
half boxes and one more (slightly smaller) box divided into two parts,
one part labeled 1/2
and the other part labeled 1/4. The drawings are
accompanied by lengthy written
These constructivist methods are inefficient, don't
generalize to more difficult problems, and give no indication that the
student knows how to set up and solve the appropriate problem in
- Later in this lesson a second problem requires
dividing 96 by 7 7/16. According to the teacher, "no one knew
what to do." So she "encouraged them to use constructivist problem-solving strategies.
She suggested that they make a model. First they cut
out a strip from used file folders that measured 7 7/16 inches long.
Then they measured and marked with masking tape 8 feet (or 96 inches)
the classroom floor. Then they "carefully measured out 12 pieces
of 7 7/16 inches each
from 96 inches" and concluded that 96
÷ 7 7/16 is "about
Fractions with Fifth Graders: Marilyn Burns first reports
that she knows about the standard way to compare fractions by
converting to a common denominator. But she wants kids to
develop their own personal ways to compare fractions. She writes:
"To help students learn to compare fractions, I used several types of
lessons. I gave students real-world problems to solve, such as sharing
cookies or comparing how much pizza different people ate, and had class
discussions about different ways to solve the problems. I gave them
experiences with manipulative materials—pattern blocks, color tiles,
Cuisenaire rods, and others— and we explored and discussed how to
represent fractional parts. I taught fraction games that required
them to compare fractions, and we shared strategies. At times I just
gave them fractions, and we discussed different ways to compare
Recall the "estimates are better than exact"
quote given above. This gives the impression that
Marilyn Burns math graduates would know how to quickly estimate the 96
÷ 7 7/16 fraction
division, but the students here don't appear to know that 8 x 12 = 96,
so they resort to hands-on methods to arrive at their estimate. This
isn't a simple problem in fraction division, but by the end of the 5th
grade the student should be able to show that the exact answer is (96
x 16)/119. If an approximation is desired, the student should
realize that an easy cancellation occurs if 119 is approximated by 12 x
10. This yields (8 x 16)/10 = 12.8 as a reasonable
Remainder of One After many examples of whole number
division (such as 9/4 and 13/6) yielding the answer 2 R1 (2 plus a
remainder of 1), the 5th grade students are asked if there is a
such that 10 divided by N equals 2 R1. Alexis "came up with the
answer of N = 9/2." Prior to this point, all examples were
limited to a whole number divided by a whole number. But Alexis
has now given a rational number (fraction) solution.
Marilyn Burns is wasting valuable time. Children need to master
the standard method for comparing fractions. It's a simple
skill, but important background knowledge, necessary for later mastery
of traditional algebra.
Crocodiles The 5th grade problem is to compute 1
+ 2 + 3 +
4 + 5 + 6 +7 +8 + 9 + 10. Jimmy said 57. Andrew and
Erin both said 55. After Kailen and Spencer both argued in
favor of 55, Jimmy caved in and agreed. All 5 now agreed.
Burns wants us to be impressed with the out of the box thinking, but
she's misleading the students and has missed an important teaching
point. The remainder concept is only necessary for the whole
number context. It's an accommodation that's needed because the
whole numbers aren't closed
under division [when you divide a whole number by a whole
number, the result may not be another whole number]. Once we
extend to the
rational numbers (fractions), we no longer need the awkward concept of
remainder. The rational numbers are closed under division. The
answer to 10 ÷ 9/2 is
correctly written as 20/9 or 2 2/9, but not 2
Marilyn Burns missed a beautiful opportunity
to teach about a famous classroom "discovery." The class just decided
that 55 was the sum of the first 10 natural numbers. What
about the sum of the first 100 natural numbers? After they
struggled with that, she could have then told them how Carl
Friedrich Gauss (1777 - 1855) solved this problem when he was their
age. His teacher was trying to keep the class busy. (Back then
they admitted it.) But Gauss quickly produced 5,050 as the
answer. He recognized that he could quickly compute twice the
( 1 + 2
+ 3 + 4
+ 5 + + . . .
+ 100) = S
2. + (100 + 99
+ 98 + 97 + 96 +
+ . . . .. + 1) = S
+ 101 +101 + 101 + 101 + + . . . . +
101 = 100 x 101 = 2S
Therefore S = (100 x 101)
÷ 2 = 50 x 101 = 5,050
Phil Mickelson Help?
To better understand the constructivist mindset, read A Summary View of NCEE Math.
- First, help your own
children. Acquire the complete set of materials (12 textbooks and
for Singapore Math Primary
Mathematics U.S. Edition. These
24 books will cost a
total of $192 + S&H. This is an excellent
the math education of Amanda,
Sophia, and Evan.
- Read Ten Myths About Math
Education and Why You Shouldn't Believe Them.
- Read The
Math Wars by David Ross.
- Seek the opinion of mathematicians employed by
- Get a copy of About
Teaching Mathematics: A K–8 Resource, by Marilyn Burns. Don't
take Bill Quirk's word for it. See for yourself.
- To better understand
how the National Council of Teachers of Mathematics (NCTM) has promoted
the constructivist mindset, read Understanding
NCTM Standards: Arithmetic is Still Missing
- Know that Marilyn Burns has a major new problem. The NCTM
recently released their Curriculum
Focal Points. Here they finally recognize the
importance of standard arithmetic.
For more examples of constructivist K-6 "problem-solving
activities," see How the
NCEE Limits Elementary School Math . Here you'll find samples
of "student-invented" computational methods.
Visit the Mathematically
Correct and NYC HOLD
Spend some time comparing About
Teaching Mathematics: A K–8 Resource, by Marilyn Burns, to Elementary
Mathematics for Teachers, by Thomas
H. Parker and Scott J. Baldridge. Which book rings
true? Which book helps you with the math education of your
- Please consult NYC HOLD's National Advisors.
We will be happy to answer your questions.
Finally, please speak out in defense of genuine math education
for American children. We know you and your wife Amy had the best
of intentions and must be shocked to discover that ExxonMobil has
appeared to endorse fuzzy math. An army of parents has also
been shocked by fuzzy math programs. These parents and their children
desperately need a champion to step up to
the tee .
Quirk is a graduate
of Dartmouth College and holds a Ph.D. in Mathematics from the New
Mexico State University.
Copyright 2007-2011 William G. Quirk, Ph.D.