Who knows?

The truth is that most Louisiana teachers have not found the Common Core standards to be developmentally appropriate, nor have they liked the preponderance of informational texts at the expense of good literature, which historically has been the source of the development of abstract thinking.

The above statement is from an article by Glynis Johnston in the The Times of Shreveport, La., on May 24 of this year.

I do not question the opinions she expresses in her article. She is entitled to them.

The phrase "most Louisiana teachers" I find truly head-scratching. If she truly has access to real data that supports her claim, she should be writing for national media and addressing her state legislature if not Congress. It is hard to fathom how these "Louisiana teachers" could have judged these standards as not developmentally appropriate at a time when very few students have tried to develop under them. No state has come even close to having students progress through their k-12 years under the influence of Common Core.  

Ms. Johnston really amazes me when she claims to know what has been the historic source for abstract thinking.  Good literature? Define that phrase. As far as I know, what is good for one is lousy for another. What I can say, based on my 36 years as a high school teacher, is that students never got enough work on reading and writing for detail. The closest they cam to reading for detail was their textbooks in math, science, and history.  The top students, however, needed to do little reading because they "got it in class" while the weak students never even took their textbooks home.  It is hard to tell the impact of reading for detail is that environment.

Perhaps these rationales for dumping on the Common Core are well-intentioned. Who knows. 

On December 1 of last year, Amanda McElfresh, writing about Louisiana, states 

Teachers also said they've seen significant improvement in student performance since they started using Common Core in their classrooms. Are these two talking about the same state?

Speaking of Ms. Johnston, back on December 5 of last year she wrote an article entitled All La. teachers do not support Common Core. I guess it is an improvement scaling back from "all" to merely "most" in her statements.

Speaking of her article's title, I believe Ms. Johnston has fallen into an increasingly common error. She uses the phrase "All...are not...", which is logically equivalent to "None... are...". I believe she intended to acknowledge that some teacher's do favor Common Core, and the phrasing "Some... are..." is logically equivalent to "Not all... are...".  In her statement the "not" appears to be misplaced.

Then again, maybe back then she really was claiming to speak for all teachers in Louisiana. Who knows...

So what is that old subtraction anyway?

Most people old enough to be parents learned subtraction in the (what I think is poor) method with "borrowing" (or "renaming" as it became known, since it was never "given back"). Their work looked something like this:

Now imagine you asked an adult to explain what is going on. Better yet, ask 10 adults. I think that the phrase "you can't take 8 from 5" and "cross out the 4 and make it a 3", or similar phrases, will be heard. You will probably not hear "50 from 30" or "change 300 to 200". Why? Because people have been trained to think of subtractions such as this in terms of a gathering of individual digits and not 2- and 3-digit numbers.

For years the phrase "number sense" was bandied about, but not enough as far as I am concerned. Generations have been trained in "digit sense" while developing little in the way of number sense.  (I use the word "training" specifically because it connotes the sense of developing a skill or talent without the need to understand why you are doing it. When was the last time you heard of someone being trained to be a PhD?)

What is happening to the numbers in the above subtraction? Here it is written out in some proper mathematical notation. (Keep in mind that once you get (got?) past arithmetic, your math work was predominantly written line-by-line, one step at a time.)

\[\begin{array}{c}345 - 158 = (300 - 100) + (40 - 50) + (5 - 8)\\ = (300 - 100) + (30 - 50) + (15 - 8)\\ = (300 - 100) + (30 - 50) + 7\\ = (200 - 100) + (130 - 50) + 7\\ = (200 - 100) + 80 + 7\\ = 100 + 80 + 7\\ = 187\end{array}\]

If we can accept for the time being this line-by-line approach as having some legitimacy, consider this:

\[\begin{array}{c}345 - 158 = 345 - 100 - 50 - 8\\ = 245 - 50 - 8\\ = 245 - 45 - 5 - 8\\ = 200 - 5 - 8\\ = 195 - 8\\ = 187\end{array}\]

I include this because it is different, perfectly correct, and more easily followed mentally

My method for doing this subtraction is like this:

\[\begin{array}{c}345 - 158 = 245 - 58\\ = 195 - 8\\ = 187\end{array}\]

By having taught myself years ago to work with numbers left-to-right, the same way we say them, I can claim that I haven't "borrowed" (or "renamed") while doing a subtraction in decades.

If all you know is the handwritten version above, odds are you will find few shortcuts and will stumble doing subtraction mentally.  If you find yourself at a checkout counter with a clerk who has trouble making change, especially when you hand over a twenty, two quarters, and a dime for your bill of $19,58, consider that the clerk, who is quite possible younger than you, has only the "borrowing" method to use, which is very very awkward to do mentally.

An ellipse is born

I thought I'd just post a sampler of what is happening on my other digital stomping grounds. This starts with two circles that can rotate around two circles with the same center. They are rotating in different directions, but their orbits take the same amount of time. The end creation is an ellipse. Below is a derivation of a Cartesian equation as an #Algebra II or #PreCalc student might do it.

The important thing here is that dynamic geometry such as #GeoGebra can open doors to concepts not normally seen until later years. The basic construction here (minus the buttons and check boxes) can be handled by a middle school student. The entire GeoGebra file can be found here.

Time to Reverse Direction

I have read a lot about people blasting Common Core for de-emphasizing, if not leaving out, the old right-to-left "borrowing" (or "renaming") method of subtraction.

To all those people I should first point out that even on a calculator, numbers are entered left-to-right. Having students learn right-to-left for paper calculations, but left-to-right for electronic calculations, has helped create a generation that finds mental arithmetic almost totally impossible.  Conflicting methods all through school create havoc in the student's brain.

Imagine learning to write right-to-left. Quickly, spell your full name backwards.

The Common Core recognizes that teaching right-to-left subtraction does not promote understanding of what subtraction really is.

All over the web there are people blasting supposed new and different techniques for subtraction. Business Insider had one last October (see it here). In it the author, Andy Kiersz, writes

The "counting up" method (which is what's depicted in the textbook above) is not intended to replace the standard way. Instead, it captures some of the underlying aspects of subtraction and place value that allow borrowing and carrying to work.

His comment bothers me. It is based on the assumption that the old methods are fine, just in need of a facelift.

In fact, they were not fine. The whole mess of right-to-left arithmetic needs to be tossed. We say numbers left-to-right, we write numbers left-to-right, we enter them on calculators left-to-right, we even say, remember, and dial our phone numbers left-to-right. Why would anybody think that we should be calculating with them right-to-left?

Another article referring to the same example is by Ed Morrissey, also from last fall (see it here). In it he says "the “counting-up method” requires a paper calculation and more complicated cognitive judgments than simply subtracting and carrying over."  That analysis is highly debatable. Anyone grounded in left-to-right addition finds the 2+60+200+25 fairly easy.  The process does not require paper calculation at all. In fact the need to write down paper calculations will diminish as one gets more familiar with left-to-right arithmetic.

Thankfully, students are beginning to learn arithmetic in a manner that will help them internalize the concept making them stronger and better thinkers. The "New Math" back in the 60's tried to do that, but it never dealt with the "right-to-left" vs "left-to-right" issue. Back then it was apparently presumed that if people obtained a better number sense the old arithmetic methods would make sense. That was still climbing the uphill battle because right-to-left was against-the-grain.

The examples referred to in Kiersz' article begin to make full sense to someone who has begun to think and work with numbers left-to-right. In the "subtract 38 from 325" question, I would subtract 40, get 285, then add 2 back to get 287.  It is different from "counting up", but it's much easier to understand than "you gotta take 8 from 5 but you can't take 8 from 5 so you take the 2 in the tens place, make it 1, and and a 10 to the 5 in the ones place making 15,.....". (Apologies to Tom Lehrer). 

Common Arithmetic

If our Common Core testing fiasco yields nothing else, I hope it changes the way arithmetic is done in this country.

Scenario: you are told to add these 4 numbers. How do you do it?

If you proceed the traditional "paper and pencil" way, here are your thoughts:

7 + 8 = 15
15 + 6 = 21
21 + 3 = 24Write down 4 under units columnWrite 2 at top of tens column.2 + 4 = 66 + 7 = 1313 + 4 = 1717 + 1 = 18Write down 8 under tens columnWrite 1 at top of hundreds column1 + 3 = 44 + 2 = 6Write down 6 under hundreds column

This took 14 steps, and at no point did you say or even think any number even close to your answer.

Now we will do it left-to-right. Here are the thoughts:

300 + 200 = 500 500 + 40 = 540 540 + 70 = 610 610 + 40 = 650 650 + 10 = 660. 660 + 7 = 667 667 + 8 = 675 675 + 6 = 681 681 +3 = 684 Write down 684

Ten steps, and at each step you were actually thinking numbers that got closer and closer to the final answer.

Why is the first option the most common that people remember? Because single digit arithmetic is the first taught and learned, and it does not vary far from that.

If we treated transportation the way we have treated arithmetic, we would all still be walking.

Will it go round in circles?

The beginnings of a Ferris Wheel done in GeoGebra. Go to here. download it, and play with it. Finish it. Change it. Explore. Learn.