Thursday, February 7, 2013

Invented Math

Question:  Will the Common Core Standards Fix This?

It’s a rare thing anymore for me to meet a student who knows how to subtract correctly.  Many of them subtract upside down, at least for part of the problem.  I am not talking about little 1st graders who are just beginning to learn;  most recently I am talking about having met a 2nd grader closing in on third grade, a 6th grader, and an 8th grader all doing the same thing.  Faced with a problem such as 92 – 18, too many of today’s students will place a 6 in the ones column and an 8 in the tens.  The 8th grader started reaching for his calculator when I told him he was wrong.  He’s been using a calculator for 3 - 4 years now and knows how to do absolutely nothing without it.  That young man opted for an after school tutoring program when he saw I wouldn’t allow him to use his calculator - nor count on his fingers.  (I’m hoping they’ll finally teach him how to do something besides push buttons.)  The 2nd grader will be fine, and the 6th grader is learning the multiplication table with mini lessons on adding & subtracting using the age-old, efficient procedures everyone grew up with before the Educators started foisting New Math on us. 

It isn’t hard to figure out how this has been happening – the standards used by Florida, and many other states, have the teachers trying to cram so much material into the kids that there is no time for them to actually learn anything.  The curriculum has everyone flying from one disconnected area to another, often spending no more than a week on any one topic, no matter how complicated, and no matter what the students’ skill set.  One week it’s the four operations with fractions, the next it’s converting decimals to percents – this with kids who hardly know what a fraction is and who have the same problems as above with the four basic operations even with whole numbers. 

Compounding the problem is a lack of emphasis on teaching the standard algorithms for arithmetic operations.  By “standard algorithms” I mean the step by step procedures we older folks all learned as kids.  For instance, with a 2-digit subtraction problem, the student will subtract the ones column first, borrowing if the bottom number is bigger than the top, then subtract the tens column.  (The Experts make a big deal out of calling borrowing and carrying “regrouping” or “exchanging” as if this clarifies matters.  I don’t care if they call it seizing and hauling as long as the kids learn how to solve the problems.) 

Instead, we have parents as well as students struggling with the results of nonsense such as Florida’s 2nd grade standard MA.2.A.2.2 which states: “Add and subtract multi-digit whole numbers through three digits with fluency by using a variety of strategies, including invented and standard algorithms and explanations of those procedures.”   The key phrases are “variety of strategies” and “invented” algorithms.  This translates to the classroom as an enormous number of methods that must be learned for even the simplest tasks, and they really do mean to have the kids try to invent their own strategies for solving math problems.  This reminds me of how they expected kids to “invent” spelling or “construct” the meaning of text during the dark days of Whole Language.  I am not kidding.  Below is just one quote that can be found by Googling math algorithms.

“Invented strategies are flexible methods of computing that vary with the
numbers and the situation. Successful use of the strategies requires that
they be understood by the one who is using them—hence, the term invented.
Strategies may be invented by a peer or the class as a whole; they may even
be suggested by the teacher. However, they must be constructed by the

The Experts must have gotten tired of waiting for the 7 year olds to come up with algorithms that took a series of geniuses a few centuries to figure out, so they decided to be creative and innovative and invent some themselves.  Under the seductive guise of helping children understand the concepts of math, it’s as if they were trying to make math as hard and horrifying as possible.  Below is one of the more counterproductive “invented algorithms” I’ve seen so far:

To solve 35 – 9, the student is supposed to cross out one of the tens and fill in the bottom block of ten ones with ten little boxes.  Then the student is to cross out the top 5 boxes in the top block of ones plus 4 in the bottom block.  This leaves 2 tens with 6 ones in the bottom block.

The next problem, which is recreated on the back of an envelope, is set up to solve 63 – 28.  The students are to cross out 3 tens, filling the bottom block of ones with ten little boxes (I used slashes).  Then the student is to cross out the 3 ones in the top block plus 5 ones in the bottom block which leaves 3 tens and 5 ones as the answer.

The above were taken from a second-grader’s worksheet with the appropriate standard (above) written in the upper right corner. (Publisher: Houghton Mifflin Harcourt)

The expanded form is popular for far too long in a student’s career.  This looks like:

15 + 39
10 + 5 + 30 + 9
10 + 30 + 5 + 9
40 + 10 + 4
50 + 4

This can go even further and look like this at a higher grade level:

34 + 27 = (3 x 10 + 4) + (2 x 10 + 7)              expanded form
= (3 x 10 + 2 x 10) + (4 + 7)                            associative and commutative properties
= (3 x 10 + 2 x 10) + 11
= (3 + 2 + 1) x 10 + 1                                      distributive property
= 6 x 10 + 1 = 61                                             simplified form

And there are these (for 3rd Grade):

     I found the paragraph under this example pretty interesting.  The kids are instructed to work in pairs or groups, hence the "shared" algorithms.  And the remark about how adults use the method that starts with the 1"s - silly grown-ups!

    And there are these:

If you would like to explore the wonderful world of invented math, all you have to do is Google something like: “Standard algorithm for addition”.  Give yourselves plenty of time, there’re just tons of these examples out there.  I clicked on a couple dozen – they would start off with showing the requested standard procedure and then go off on their flights of fancy.  Many of the sites that exhibit alternative algorithms for basic math have university addresses and are put up by folks who no doubt consider themselves the “Experts” in their fields.  This is not a reason to believe that their opinions deserve any special kind of respect, especially since all their theorizing is proving to be such a dismal failure for so many children.  Should you feel yourself being swept up in their seemingly high-minded, intellectual sounding, jargonistic gobbledy-gook, just remember, it’s people just like these who brought our children travesties like Whole Language and invented spelling – indeed the entire dumbing-down era of OBE and Blueprint 2000.

The public schools are confusing the introductory modeling of math concepts with the methods for performing the functions, and the resulting mish-mash isn’t working.  In fact it’s proving to be detrimental to the academic progress of American students.  This is why we keep having to import so many foreign born, and educated, students to fill up the higher education seats in American Universities – we’re not even teaching our kids how to add and subtract correctly.  And the jobs – the high end, math dependent jobs.

And guess what?  Singapore schools teach their students the standard algorithms!  Duh!

So – are the Common Core Standards going to fix this?  Maybe not.  We’ll be taking a close look.