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 2 of this series of posts: the pedagogical reasons for using an open number line.

I firmly reject the premise, implied by the picture, that the only goal of teaching mathematics is just getting the right answer. If that’s the goal, then we can eliminate much of the K-5 curriculum by handing out 19-cent solar calculators to every student. That will pretty much guarantee that the students will solve elementary arithmetic problems like the one shown. It will also fail to develop their higher-order reasoning skills and leave them hopelessly unprepared for algebra.

A great example of a teacher using children’s natural curiosity to foster mathematical thinking can be seen in this video: https://www.teachingchannel.org/videos/common-core-teaching-division.

I absolutely agree with Common Core critics that students in elementary school should, at the end of the day, be able to quickly compute 427-316 using the standard algorithm in 5 seconds or less. I would also argue that it’s also important for students to learn why is algorithm works and not just how to use the algorithm. This deeper level of conceptual understanding begins to lay the foundation for the more abstract thinking required in later years.

The open number line predates the Common Core by about 25 years; for more information, see http://www.k-5mathteachingresources.com/empty-number-line.html. When used properly (a big “if”), it will lead students to the standard algorithms for addition and subtraction. I’ve personally seen the open number line successfully used in the elementary schools where I live in the hands of skilled teachers. Bottom line: the students had no problem with the technique. (By the way, Texas is not a Common Core state.)

So I take great issue with the quote “The process used is ridiculous and would result in termination if used.” Of course the process is ridiculous for people who already know how to subtract. For students who are first learning how to subtract multi-digit numbers, however, it is a completely appropriate way to introduce students to the topic which will lead them to the standard algorithm for subtraction. Once learned, then there will be no need to revert back to the previous method.

I have heard some critics say that teaching the “why”s of mathematics, using the open number line and other techniques, merely favors the stronger math students and the expense of the weaker students. Perhaps this happens in some classrooms. However, this has not been my personal experience. I have supervisory authority over aspiring teachers who have taught math to literally thousands of third, fourth, and fifth graders — all in a non-Common Core state. With careful supervision, they have used the children’s natural curiosity and inductive reasoning (but not deductive logic — that’s an important distinction) to get at the “why”s of mathematics as well as the “how”s. And I’ve not personally witnessed anything like the frustrations that I’ve heard as the Common Core has been implemented. Hence my “don’t throw out the baby with the bathwater” approach when I hear and appreciate legitimate concerns and frustrations with how the Common Core has been rolled out.

One more thing: a great motivation of the Common Core has been to improve student achievement in algebra by fostering mathematical habits of thinking in the lower grade levels. It’s certainly debatable about whether or not this ideal has been achieved. But the motivation for the Common Core and improving conceptual understanding at the lower grade levels is to increase access to higher-level mathematics, not to favor the high achievers.

Though I’ve seen it used effectively, I’m personally not an advocate for the open number line. Given the choice, I would use a base-10 kit (shown above) for students who are first learning subtraction. I would start by collecting 4 plates of 100, 2 rods of 10, and 7 cubes of 1. I would then remove 3 plates of 100, 1 rod of 10, and 6 cubes of 1. The remaining pieces (1 plate of 100, 1 rod of 10, 1 cube of 1) represents the answer of 111.(This particular problem didn’t require trading a plate of 100 for 10 rods of 10 or else a rod of 10 for 10 cubes of 1; this would be handled later in the curriculum.)

I do recognize that teaching children multiplication and especially division with a base-10 kit is very cumbersome and is significantly easier using an open number line. So I see the rationale for using an open number line for addition and subtraction if it will be used in later years to teach multiplication and division.

I don’t support requiring children to learn subtraction or any other concept in mathematics by any one specific technique. The big issues are (1) getting the right answer and, of equal importance, (2) understanding why the answer works. I take issue with the original picture because it favors (1) but not also (2), and a surface level of understanding will not serve students well when they hit algebra.

Bottom line: I have absolutely no problem with schools using a technique like the open number line, as it is a mathematically correct way of explaining to students why the answer works.