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.


  1. Love your frustrating when there is so much misinformation out there. I wish I had had some of these strategies intoduced when I was in school. I was an adult before I figured out why we 'carried and borrowed.' I agree with you - what parent wouldn't want their child to have this understanding.

    1. Sometimes it's hard to sort through all of the misinformation! Other than counting up to count change back to customers, I also learned many of these strategies as an adult in the math methods courses I took to become a teacher.

  2. This comment has been removed by the author.

    1. Reply in haste - repent at leisure! Here's a revision of the comment I removed:

      Here's my concern: "Counting up" to benchmarks like the nearest 5 or 10 is a great strategy for a question like 100-78, but not for 32-12. I would not likely try to represent 32-12 using a counting up/benchmark strategy - your first open number line is not that much clearer than the vertical method in the Facebook image (and this is coming from someone who LOVES open number lines!) Parents go crazy when we demonstrate methods like these for questions that should be pretty straightforward, and could be solved in much less complicated ways. I worry that this is driving the push-back against math reform, and the longing for "the good ol' days."
      I'm not sure teachers always begin with the ideas of their students: "How would you find this answer?" Subtract the 10 and then the 2? Good idea! Subtract the 2 first? Why not! Count up by 10s? Great! If some student suggests adding up from the 12 to 15, to 20..., let them demonstrate that, and compare it to the other strategies suggested by other students. Which method is easiest with these numbers? Which would you choose to use? Why?
      We're aiming for flexibility. The process needs to be student-centered - driven by the ideas of the students. So many textbooks and other resources fall back on explicit instruction by the teacher. We need to ask questions designed to push students to suggest the strategies we're looking for (never say anything a kid could say for you!) and then help them explore those strategies, and record them in efficient ways, and think about where and when to use them.

  3. You are absolutely right that we need to begin with the ideas of students, that the process needs to be student-centered, and that we need to teach them to be flexible, strategic, and efficient--I love your description!

    While I agree completely with your comments about how we should be teaching I have a feeling you may have misunderstood the purpose of my post. Whoever created the "Old fashion" way vs. the "New" way problem that was posted to Facebook chose the numbers 32 - 12 and chose the "counting up" strategy to demonstrate the "New" way with stacked equations.

    My purpose here was to make the following points in direct response to the photo (using the same numbers used in the Facebook photo):

    1) Counting up is not a new strategy & is used to count back change,
    2) The open number line was the best way I could think of to visually represent the strategy used in the "New" way (not to imply I would teach this problem in that way, or that this is the only other possible representation),
    3) The "New" way has nothing to do with the Common Core--the CC doesn't prescribe that students use the strategy in the photo (or any particular strategy for that matter), and
    4) Students need to develop number sense, have a flexible, conceptual understanding of addition & subtraction, and use the mathematical practices listed in the box.

    I appreciate your point "parents go crazy when we demonstrate methods like this . . ." so I'm going to edit my post so it's clear that I'm not suggesting that the problem should be solved by using counting up or with an open number line.

  4. Elizabeth in New EnglandOctober 8, 2014 at 8:09 AM

    There are myriad issues with the Common Core... while I appreciate this post, I took issue with two parts of it:

    1. You point out grade-level appropriateness of some of the work, but then write it will be worked out over time.

    My children are suffering *right now* with material that is inappropriate for their grade level. In addition, my school system has acknowledged that following CCSS has created "gaps" in learning, whereas kids in 5th grade who might have been learning 5th grade material, are now learning higher level material. There is currently no plan to address these gaps in knowledge. This generation of kids are being wholesale (across the country) being used as test subjects for someone's theories on education. I'm just not cool with that.

    2. I would love to see some research showing that the CCSS prescribe (because, although you do give the standard line that they are not a curriculum, they do drive curriculum, and are making a lot of money for Pearson, et. al). I read this yesterday... do you have any comment?

    P.S. - I also have huge issues with CCSS ELA - my children are reading/learning about world religions in K, first, and second grade instead of learning about their own community. Further, my daughter is in second grade - they have already finished their ONE UNIT that will include Fiction. All other work this year is non-fiction. Students are also graded only on the author's intent, not on what feeling the writing incites. What if Lincoln's intent for the Gettysburg Address was much different from the reaction it engendered? It wouldn't matter under CCSS. There's much more to say, but I'll stop for now.

  5. Hi Elizabeth, thanks for your comment! You've said a lot there!

    My purpose with this blog post was to illustrate that the math method (on the legal pad paper) IS NOT CCSS-M. If that is actually how a teacher were teaching this, then there is a much bigger problem than the CCSS-M.

    I heartily agree with you on many of your points--schools and children should not be used by big business to make big money. I am completely opposed the inappropriate use of standardized testing (see my post "Teens With an Ax to Grind")-- students should not be used as test subjects and the scores should not be used for teacher evaluation (the subject of my as yet unfinished dissertation). I have huge issues with the CCSS-ELA-- the over emphasis on nonfiction and I am especially concerned about the "Close Reading". It is one way to read, not 'the' way to read.

    With that said, I do give the "standard line" that the CCSS-M are not a curriculum--is in fact a direct quote from the CCSS-M Introduction>>How to read the Grade Level Standards:

    "These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B." (

    In my opinion school districts (and teachers, for that matter) that choose to ignore this statement are committing educational malpractice. I drive this point home in every professional development session that I deliver. However, the issue then becomes implementation, not CCSS. Unfortunately, implementation is a big problem with any new initiative.

    As far as research goes, I recommend that you read Dr. Jo Boaler's "What's Math Got To Do With It?" and check out this teacher's testimony:

    I hope you'll keep advocating for children!