The Fuzzy Math Mindset Behind Phil Mickelson ExxonMobil Math

by William G. Quirk, Ph.D.   (E-Mail: wgquirk@wgquirk.com)   

Each summer, elementary school teachers attend Phil Mickelson ExxonMobil Teachers 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 ExxonMobil Teachers 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 math education. 

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?

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 build 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 "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.  

What Should Children Learn in Elementary Math?

The Marilyn Burns View of How Children Should Learn "Math"

Marilyn Burns has published several 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 a child's 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."

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."  

Marilyn Burns Rejects Standard Paper-and-Pencil Arithmetic

Marilyn Burns is vague about specific math learning objectives.   But she's clear that children shouldn't learn standard arithmetic.  Here are quotes from  Teaching Arithmetic, Part III of in About Teaching Mathematics: A K–8 Resource.   
  1. "Facility with standard paper-and-pencil arithmetic is no longer the measure of arithmetic understanding and competence."    Page 139
  2. "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 
  3. "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
  4. "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
  5. "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
  6. "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
  7. "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
  8. "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].
  9. 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."  
The X-O Problem   This 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 at 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 was 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."
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. 

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. 

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.  

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 + 12H2 = 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. 

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.
  1. A  Fractions Lesson 
  2. 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."  
  3. 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. 
  4. 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. 
 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

How Can Phil Mickelson Help?

  1. 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.
  2. Read Ten Myths About Math Education and Why You Shouldn't Believe Them.  
  3. Read The Math Wars by David Ross.  
  4. Seek the opinion of mathematicians employed by ExxonMobil.   
  5. 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. 
  6. 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
  7. 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.
  8. To better understand the constructivist mindset, read A Summary View of NCEE Math. 
  9. 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. 
  10. Visit the Mathematically Correct and  NYC HOLD websites:
  11. 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?
  12. 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 .
Bill 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.