The following picture has been making the rounds lately.

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

- The solution of this particular question.
- The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
- The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
- The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

This is Part 1 of this series of posts: the solution of this particular question. Here it is:

Jack correctly started at 427 on the number line. He then correctly understood that 316 consists of 3 groups of 100, 1 group of 10, and 6 groups of 1. He then correctly subtracted 3 groups of 100 (for an interim answer of 127) and then correctly subtracted 6. However, he forgot to subtract 10. That’s why he got a wrong answer (121) that was 10 more than the correct answer (111).

Just to make sure I wasn’t completely missing the mark on this, I rewrote the problem (without the handwritten commentary) and showed it individually to a few elementary school students. They all saw Jack’s mistake within 15 seconds. They may not have been able to explain what Jack did right and what Jack did wrong in the form of a letter (more on that in a later post), but they certainly identified the core problem quickly.

I understand a parent’s frustration with knowing how to subtraction but seeing a child learning subtraction in a different way. (More on that in a later post). I also understand that some may argue with this technique of teaching children how to subtract. (More on that in a later post.) But there’s no way to sugarcoat this: an engineer who took differential equations and read this problem but couldn’t figure out that Jack forgot to subtract by 10 has little conceptual understanding of mathematics.

So let offer some free advice to critics of the Common Core who want to share this picture to vent their complaints. I am totally sympathetic with frustrations expressed in this picture. Sharing this picture with your fellow critics may feel good, perhaps with the self-justification “If an engineer can’t figure this stuff out, then how can I?!?!” However, sharing this picture is not going to persuade anyone who disagrees with you to your cause. Remember: some children can solve this problem in 15 seconds or less. If anything, sharing this picture only communicates to those who disagree with you that the critics of the Common Core are the people who have little conceptual understanding of elementary school mathematics. Once again, I am sympathetic to the emotions expressed in this picture, but there are better ways of criticizing the Common Core and persuading its unabashed supporters to your cause.

In the posts that follow, I will provide plenty of criticism of how the Common Core has been implemented in its initial years.

## hogamy

/ April 11, 2014It’s funny – one of my reactions was that this picture illustrates how educated people who should know math actually have little conceptual understanding of fundamentals. If I were ‘frustrated parent’, I’d be embarrassed to share that. I’ve never heard of open number line (that I recall) but I could figure this out in less than 30 seconds – I’m an engineer too and it wasn’t hard to figure out at all. If it were taught to me, I’d probably be faster and it’s posts like this that are missleading and spread fear/anger vs logical analysis of what’s going on.