New Math – Creative Estimation

Ever since the 90’s, when Outcome-Based goals were adopted by just about every public school in the United States, one of the most frequent complaints about Fuzzy Math has been the over-emphasis on estimating.  Along with insisting that students try to calculate answers using a multitude of convoluted, ultimately baffling methods, this is another way time is taken away from practicing actually solving mathematical problems with the standard algorithms.  Teaching children the familiar method of rounding to various place values, however, has taken a back seat to “creative and innovative” contrivances invented by the Experts. 

One of the “new math” estimation methods that recently came to my attention is the use of “compatible numbers”.  The following examples came from the 6^th Grade text book titled Big Ideas Math (Florida Edition), copywrite 2010.  One author is Ron Larson, whose Bio includes all kinds of high-faluting accolades, including a Ph.D. in mathematics from the University of Colorado in 1970.  The second author is Laurie Boswell who teaches math in Vermont, has received the Presidential Award for Excellence in Mathematics, and – most important – has served on the NCTM Board of Directors.  The latter  (National Council of Teachers of Mathematics) is the group of alleged math experts who, among other things, don’t believe in paper and pencil exercises, have effectively done away with teaching long division, and continue to push doing even the most basic computations on calculators.  I’ve included all this information about the people who are at least in part responsible for the state of math skills in this country to show that these folks are considered at the very top of their field.  It is too easy for parents and others to become so cowed by all the degrees and awards and honors, etc. that they feel their concerns must be baseless.  What is happening in our public school classrooms, however, warrants a serious look beyond the professional glitz.  Here are some examples of what these Experts have our kids “learning”.  (I've included photos because my typed fractions are confusing.)

First the students need to know their fractions really well (my students do not).  In the first example of estimating products of fractions, 3/8 rounds up to 1/2, and 11/12 rounds up to 1 so 3/8 x 11/12 ≈ 1/2 (11/32).  In the next, 4/5 rounds up to 1, and 1/6 rounds down to 0 so 4/5 x 1/6 ≈ 0 (2/15).  

In estimating products or quotients of mixed numbers, 5 ¼ rounds down to 5, and 3 9/10 rounds up to     4 so 5 ¼ x 3 9/10 ≈ 20 (20 19/40).  In the second example given for this type of problem, 11 5/6 rounds up to 12, and 2 2/3 rounds up to 3 so 11 5/6 ÷2 2/3 ≈ 4 (4 7/16).  (I have no idea why there are no examples of estimating the division of fractions.)  (These are all found on pg. 46)

Okay.  I have no memory of estimating fractions myself, but I actually have no objection to learning the comparable sizes of them well enough to do so.  Many of today’s students do not have this solid grounding, however.  Then, to make matters even more complex, the authors introduce “compatible numbers” to the mix.

“Compatible numbers are numbers that are easy to compute mentally.” (pg. 47)

For solving 275 ÷ 3 ¾, first we round the 3 ¾ up to 4.  Then we realize that 4 does not divide evenly into 275 – so we change it.  If somehow the students have grasped how to round mixed numbers, this is where many of them will fall flat.  They are assumed to know that they can change 275 to 280 which is much more convenient for dividing by 4.  275 ÷ 3 ¾  ≈ 70 (73 1/3).  (Please understand that most of the students who come to me needing help with this kind of work do not know their multiplication tables.)

This was totally new to me, so I worked several problems (pg. 48) to see if I got them right before trying to guide one of my students through them.  As it with so many of today’s Math text books, half of the answers are in the back of the book – all the odd numbered problems in this case.  I was doing pretty well until I got to ¾ x 1/3 which I estimated as 0 (1 x 0), but the answer in the book is ½.  Then I got to 48 ÷ 6 7/12.  First I rounded the 6 7/12 up to 7 & then changed 48 to the more convenient 49 and came up with 7.  The estimated answer given in the book is 8. The actual answer is 7 5/13.

Is it any wonder?

And another interesting tidbit – this stuff is NOT on the FCAT.

Current Events - Florida Test Errors

A mom I know who has children in Florida’s Hillsborough County public schools was telling a group of us about her daughter coming home and complaining about all the grammar errors on a Language Arts test she had taken that day.  This was a Hillsborough County test, not the FCAT.  This 6^th grader counted seventeen errors in the text of the materials themselves; these were not for the students to identify and correct errors deliberately inserted as part of the test.  These tests may have been part of the effort to design the course-specific evaluations that are to assess the new Core Curriculum standards that have the entire nation in a tizzy.  Or perhaps they are more connected to the teacher evaluation effort financed by Bill Gates’ millions.  Hillsborough was one of the districts chosen by the Gates Foundation to pilot school improvement ideas in return for piles of money.  If it’s the latter, we would have to wonder how happy Mr. Gates would be about the lack of quality in such important assessments.

Parents have become too used to finding errors in the written work of Educators at all levels.  Teachers are often lax in spelling and grammar, especially in letters and notes to parents.  This is true even when spelling and grammar checking are readily available to them.  At a slightly higher level, at which tests are designed and written, we have come to expect no better. (Kind of like what we expect of newspapers these days – doesn’t anyone employ editors anymore?)

Take a look at the second grade Reading Comprehension test below.  It looks a bit uneven because the last couple sentences were on a different page.  This multiple choice test was given on a computer at a Pasco County Elementary school using a program called “ExamView Web Site”.  The student had scored poorly, and the concerned parent wanted to see the kind of questions her child had missed.  Ordinarily, parents receive only the question number, the letter response of the student and an “X” next to the errors – not the narrative, not the questions, and nothing very helpful to the worried parent. 

Notice that the first paragraph is indented, but the rest are not.  For children who have been taught that paragraphs start with indented lines, this would be very misleading.  In fact, this child took the test two times, and both times missed number 5 which read, “Which sentence below tells the main idea of the SECOND paragraph?”  There were 11 multiple choice questions plus an essay question, yet the instructions were to “answer the question”. (Sic)  The scoring notes at the top of the test stated that the essay question would not be graded, but that was only half true.  The content was graded, but the spelling and grammar errors were not marked nor noted in any way.  Apparently the scourge of invented spelling is still with us from the days of OBE/Blueprint 2000.

If this parent had not asked to see the actual test, these errors would not have come to light.  The test given to the 6^th grader in Hillsborough is never even seen by the students’ teachers, let alone by the students’ parents.  This is a serious down side to the computerization of the public schools – it can completely obliterate the tiny bit of accountability to the parents that still exists after decades of shoving parents aside.  This takes us in the opposite direction of what is needed if we are to turn our public schools around.  We need more accountability – to the parents – not less.

These examples of sloppiness in the quality of school assessments show a deep lack of respect for students, their parents, teachers, and for the educational profession itself.  It shows a grudging, half-hearted effort on the part of Hillsborough schools in making worthwhile use of Bill Gates’ good-faith financial support for improving student academic achievement.  And it also shows a critical absence of professionalism from people who keep demanding they be treated as professionals.