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 3 of this series of posts: assessing the depth of a student’s knowledge of mathematics in ways that are developmentally appropriate. To be blunt, this has been one of the great weaknesses of the roll-out of the Common Core in the early years of its implementation.

Where I agree with most critics of the Common Core is that, even though I want students to understand why (for instance) subtraction works, 2nd and 3rd graders are still learning how to write complete sentences. So of course they will get easily frustrated with explaining their reasoning in paragraph form. I think there are better ways of assessing whether young children really understand subtraction that is more developmentally appropriate. (Here in Texas, students in this age group are instead asked to explain their reasoning by drawing pictures.)

I’m happy to defend the standards of the Common Core themselves, but it’s patently obvious that the implementation of the Common Core standards were driven by textbook publishers and not educators. I don’t doubt that much of what’s assigned for homework under the guise of the Common Core is developmentally inappropriate.

I’m perfectly happen to defend the logic behind a specific pedagogical technique like the open number line. But I can still be genuinely annoyed that the first years of the Common Core has been so badly botched. And I’m especially annoyed that most public defenders of the Common Core seem deaf to the legitimate and thoughtful complaints of its critics (as opposed to the braying expressed in the original picture).

Here’s another example that made the rounds in recent months, for which I have a considerable amount of sympathy.

The problem in question that forms the basis of her argument:

Mr. Yumata’s class has 18 students. If the class counts around by a number and ends with 90, what number did they count by?

The “correct” answer requires students to divide 90 tick marks into equal groups of 18, counting the number of tick marks in each group. Again, the textbook publisher missed the mark. The phrasing of the problem asks for just the answer (5). If the problem wants to know a rationale for the answer, then it should have been phrased in a different way.

There absolutely is a place in the curriculum for the method described in the video — dividing objects into piles of to see how many objects are in each pile. This works well when children are *first* exposed to division and and are reasonably small integers. However, by the time students get to and , another method should be used. By this point, dividing objects into piles is logically correct but pedagogically questionable, as it requires precisely 108 steps (as noted in the video). Any little careless mistake in counting will lead to an incorrect answer.

Flatly, I won’t defend the indefensible way that the Common Core has been rolled out. The textbook publishers have clearly missed the mark on how to assess the depth of a student’s knowledge. Voters will be more than justified in voting out anyone who supports the Common Core if its implementation isn’t fixed in the very near future.

Paragraphs at this age is a little much. The better way to evaluate may be to solve the problem in the correct way. Just keep it simple…I really hope the implementation is fixed soon too because these standards really aren’t the evil beast that everyone makes them out to be.

I have a fair amount of sympathy for the Electrical Engineer Dad in the first picture. Beyond the issue of whether the student should be able to write the paragraph explaining her findings, the student isn’t just being asked to apply the open number line to solve 427 – 316. They are being asked to figure out where the fictional Jack made his mistake. In order to do that detective work, they need to observe and recognize the significance of the numbers ‘Jack’ has written – that he went down by 100’s to 127, then down by 1’s to 121 – ah, Jack has skipped the step of going down by 10 from 127 to 117. That took me a while to figure out myself – partly because all of the down by 1’s steps look about 1/10th the size of the down by 100’s steps – so like Engineer Dad I initially thought Jack was counting down by 6 10’s from 127 to 67. It seems cruel to subject the young learners this is intended for to deceptive visual cues like that.

I also find myself wondering if the open number line idea itself is pedagogically sound. It logically sound of course, but does it undermine the understanding of subtraction as the inverse of addition that the traditional number line builds (addition = move to the right, subtraction = move to the left.) Do you have any opinion on that?

Don, thanks for writing.

To your first point, I’ll be the first to admit that I’m not an expert on childhood development. However, my personal opinion is that, as a general principle, identifying a mistake is a developmentally appropriate way of assessing the depth of a elementary school student’s understanding.

I do agree that the step size of the 100s look about ten times bigger than the size of the 1s. For its advocates, that’s a strength of the open-number line: the scaling doesn’t have to be perfect for students to arrive at the correct answer, and I’ll (perhaps charitably) assume that students were familiar with these scales with in-class practice problems and were asked to do other examples for homework.

To your second point, I believe that the open number line is pedagogically sound (even if I’m not an advocate for this method). Please see my previous post https://meangreenmath.com/2014/04/09/common-core-subtraction-and-the-open-number-line-part-2/ and also the link http://www.k-5mathteachingresources.com/empty-number-line.html contained within.