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 6th 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.