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 learn today. Approximately 80% of K-6 math education should still be devoted to the mastery of standard paper-and-pencil 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 college mathematics. 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 this paper.

- How Should Children Learn Elementary Math?
- What Should Children Learn in Elementary Math?
- The Marilyn Burns View of How Children Should Learn "Math"
- Marilyn Burns Rejects Standard Paper-and-Pencil Arithmetic
- How Can Phil
Mickelson Help?

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 "personal" is better than "standard." This may be true for arts and crafts, but standards and conventions are essential for mutual understanding and effective 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 reasoning. Remembered math knowledge is pulled together, connected in the mind, and applied to solve the current problem. External knowledge "look ups" may be involved, but remembered knowledge provides the necessary orienting framework of background information. Remembered background knowledge leads to the recognition of what should be looked up, and it leads to the recognition of how the looked-up knowledge can used. Understanding 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.

- Automatic recall of
single-digit
number facts.

- 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
a series
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].
The standard
algorithm for multidigit addition is then used to add the two partial
products, 390 and 2,340. This addition is carried out using
two
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
lower levels.

- 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.

- 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
a series
of single digit facts.
- Mastery
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
conscious thought.
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?

- Geometry:
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
end of
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.

- Calculators:
If a student must rely on a calculator for arithmetic, then later
mastery
of genuine algebra is ruled out.

- For more specific
guidance
as to what a child should know at a specific grade level, see the placement tests
at Singaporemath.com.
For the best available program for elementary math, see Primary
Mathematics, U.S. Edition.

Teacher-guided instruction ("teaching-by-telling") , step-by-step development, immediate feedback and correction, and knowledge retention techniques are all strongly discouraged. It's expected that all necessary 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 "look for 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 model."

- "Facility with standard paper-and-pencil 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."
Page 142

- "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 183
- "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
focus is
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
arriving
at estimates and being able to explain why they're
reasonable." Page 232 [Bold emphasis added].

- Comment:
Mastery of exact computations with
fractions is the number one predictor of later success in algebra.

- 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."

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
activities supposedly
include explorations of advanced topics such as algebra, probability
and statistics. But this is an obvious sham, because math
is a
vertically-structured knowledge
domain. Algebra, probability and statistics are ruled out, if the
student
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
"Problem-Solving Activities"

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-solving
activities."

A Tangram Problem Explain 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 Solutions instructors. They were to help her solve this problem. 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!

Squaring Up 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."

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 tries 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 to get the box of maximum volume." She quit!

#### Lessons
from the Marilyn Burns
Website

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.
So,
hoping that constructivist zealots won't notice the contradiction,
Marilyn
Burns promotes whole class
model lessons. Here
are four 5th
grade examples from her website.

### How Can
Phil Mickelson Help?

**Copyright 2007-2011 William G. Quirk, Ph.D**.

- These activities aren't accompanied by a statement of appropriate grade level, prerequisite math knowledge, or math learning objectives.
- 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
- Question:
What's the point? This question needs to be
asked for every Marilyn Burns "problem-solving
activity."

- 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 in Mathematical Discussions, Part 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
Problem.

Comment 2: Marilyn 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 probability.

Comment 2: Marilyn 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 probability.

A Tangram Problem Explain 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 Solutions instructors. They were to help her solve this problem. 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!

Comment 1:
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
gives
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.

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.

Squaring Up 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 recommends multiple "investigations," so she doesn't appear to know that one counterexample suffices to prove that the conjecture is false.

Comment 2: Marilyn Burns recommends multiple "investigations," so she doesn't appear to know that one counterexample suffices to prove that the conjecture is false.

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 tries 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 to get the box of maximum volume." She quit!

Comment 1:
Quitting
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
first
derivative V' = 400 - 160H + 12H^{2} = 4 (10 - 3H) (10 -
H). So
the maximum occurs when H = 3 1/3.

Comment 2: 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.

Comment 2: 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.

- A
Fractions Lesson

- The first problem is to separate a 5 3/4 pound bag of candy
into 1/2
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
formal
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
two
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
explanations.

- Comment:
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
fraction division.

- 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)
on
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
12."

- Comment:
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
approximation.

- Comparing
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
them."

- Comment:
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.

- A
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
number N
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.

- Comment:
Marilyn
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
R1.

- Counting
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.
Case closed.

- Comment: 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 desired sum:

1.
( 1 + 2
+ 3 + 4
+ 5 + + . . .
+ 100) = S

2. +__ (100 + 99
+ 98 + 97 + 96 +
+ . . . .. + 1) = S__

3. 101 + 101 +101 + 101 + 101 + + . . . . + 101 = 100 x 101 = 2S

Therefore S = (100 x 101) ÷ 2 = 50 x 101 = 5,050

2. +

3. 101 + 101 +101 + 101 + 101 + + . . . . + 101 = 100 x 101 = 2S

Therefore S = (100 x 101) ÷ 2 = 50 x 101 = 5,050

- First, help your own children. Acquire the complete set of materials (12 textbooks and 12 workbooks) for Singapore Math Primary Mathematics U.S. Edition. These 24 books will cost a total of $192 + S&H. This is an excellent investment for 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
ExxonMobil.

- 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 the Revised 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.
- See related background
information: Finding
Common Ground in K-12 Math Education.

- To better understand the constructivist mindset, read A Summary View of NCEE Math.

- 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
websites:

- Please consult NYC HOLD's National Advisors.
We will be happy to answer your questions.

- 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
three children?

- 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 .