Saturday, March 22, 2014

You've Probably Seen This on Facebook

"Old fashion" way vs. the "New" way
Fear, Uncertainty, and Doubt
UPDATED 9-19-16 See other math educator's explanations for this and similar photos here, here, here, here, & here. Also check out this link for an explanation regarding the
5 x 3 homework problem that was marked wrong.

The photo on the right has been all over Facebook the past few weeks, posted by folks opposed to the Common Core State Standards. It really is a very simple and clever way to really scare people and attempt to make the Common Core State Standards for Mathematics look really confusing--even ridiculous! In fact, it's an excellent example of the Fear, Uncertainty, and Doubt (FUD) strategy used in the corporate world and in politics to spread negative or disinformation in order to influence and/or coerce others.
 So, what's going on here?

The "New" way is actually not new at all. It is, however, a really terrible and confusing representation of a very old method (note the copyright of 1916) that's also called the "counting up", "shopkeeper" or "Austrian" method of solving a subtraction problem by addition, or counting up to find the difference. Indeed, it's how I learned to count change back to customers when I worked in my father's pharmacy a very long time ago.

Is there a better representation or mathematical model?

The creators of the above photo chose to use a confusing representation or model of this method. I know my eye was quickly drawn to the numbers in the vertical box and it took me a bit to refocus on the actual horizontal equations before it made sense to me. The open number line model below provides a visual model that's easier for me to understand the equations in the photo, using the same numbers and counting up strategy used in the photo:

In the above open number line model, I've shown the strategy of counting up to the landmark numbers 15, 20, and 30, then on to 32. This is is how I might have counted back change to a customer as a novice. Students modeling how to use landmark or "friendly" numbers (5s and 10s) might show their thinking much like this.

In the open number line model below, I've illustrated the same problem using  a "jumping tens" strategy. This is how I would have counted change back to a customer as I became more experienced. Students might also model this way as their conceptual understandings develop.

UPDATE 3-27-14: I don't mean to imply that students should be taught or required to solve  the problem 32 - 12 by counting up or by using the open number line, nor should they be required to solve the problem using the equations in the "New" way. There are more efficient strategies that could be used for "32 - 12" and  models other than the open number line that could be used to create a visual representation for the "New" way.

Common Core or FUD?

The mathematics represented in the "Old fashion" way vs. the "New" way photo above have nothing to do with the Common Core. The Common Core does not prescribe that students use the method depicted in the "Old fashion" way photo, nor does it require that the strategy or counting up be taught. The Common Core is not a curriculum and does not require any particular teaching style or method. The Common Core does require students to have a flexible and conceptual understanding of the relationships between numbers and operations before they apply the "traditional algorithms" (also known as the "Old fashion" way).  Along with the Common Core math content standards, the CC also seeks to develop positive mathematical dispositions through the Standards for Mathematical Practice, below.

What parent would not want their child to have well established number sense and a solid conceptual understanding of addition and subtraction, including strategies such as "counting up"? Don't we want all students to develop Mathematical Practices like those in the illustration on the left?

To be sure, there are some concerns with the Common Core. I worry that administrators, curriculum specialists, and teachers will not receive the professional development needed in order to effectively implement them. I worry that some standards may be in the wrong grade level, but believe that will get worked out in due time. I also worry about the excessive high stakes testing that is consuming valuable instruction and learning time, the results of which are being used to inappropriately evaluate teachers--even though it's not really a part of the Common Core.

But, don't fall for the FUD. Be informed, read about the Common Core State Standards for Mathematics. Click on "download the standards" and read them for yourself. If you see something you don't understand, ask your child's teacher or someone who does. It's highly likely someone is trying to scare you.

Please post your comments, I'd love to hear what others think about this.