Resources mentioned in this episode:

William Brownell’s article: Meaning and Skill – Maintaining the Balance…get a Free 30-Day Trial Membership to NCTM in order to access the article

Building Math Fluency video trainings 

Welcome fellow Recovering Traditionalists to Episode 58.  Today we are discussing Meaningful Mathematics.

As you may have heard in previous episodes, I know that every teacher is in a different situation right now.  You are all in need of different things, so some of these episodes may be helpful right now and some you may want to save for later when you have some down time to start planning something besides digital learning.

In today’s episode I’d like to read a portion of an article that comes from the National Council of Teachers of Mathematics.  Normally I only link to articles on NCTM’s site that you can access for free. This article is not available for free but this article is just too good not to talk about…plus NCTM is doing a free trial of their membership right now.   When you do the free trial make sure to select the option to receive access to the online archive of the Teaching Children’s Mathematics journal, that’s the journal that this article was published in.

On pages 312-313, William Brownell writes:

“There is ample evidence in psychological research on learning that the effects of understanding are cumulative. There is also ample evidence, if not in arithmetic, then in other types of learning, that the greater the degree of understanding, the less the amount of practice necessary to promote and to fix learning. If these truths are sound—and I think they are—then they should hold in the field of arithmetical learning. It follows that computational skills among school children would be greater than they are if we really taught them to understand what they learn.

Again I remind you that meaningful arithmetic, as this phrase is commonly used, is a newcomer in educational thought. Many teachers, trained in instructional procedures suitable, say, to a view of arithmetic as a tool or a drill subject, find it difficult to comprehend fully what meaningful arithmetic is and what it implies for the direction of learning. Others than myself, I am sure, have, in conferences with teachers, been somewhat surprised to note that some of them are unfamiliar with major ideas in this conception and with methods of instruction adapted thereto.

Perhaps the commonest instructional error is, in a different context, the same one that has always distorted learning in arithmetic, namely, the acceptance of memorized responses in place of insistence upon understanding. Mathematical relationships, principles, and generalizations are couched in language. For example, the relationship between a given set of addends and their sum is expressed verbally in some such way as: “The order of the numbers to be added does not change the sum.” It is about as easy for a child to master this statement by rote memorization as to master the number fact, 8 – 7 = 1, and the temptation is to be satisfied when children can repeat the words of the generalization verbatim. Similarly, the rationale of computation in examples such as: 33 + 48 and 71 – 16, makes use of concepts deriving from our number system and our notions of place value. But many a child glibly uses the language of “tens” and “ones” with no real comprehension of what he is saying. Such learning is a waste of time. To use an Irish bull, the meanings have no meaning.

I intend no criticism of teachers. Until recently there have been few professional books of high quality to set forth the mathematics of arithmetic and to describe the kind of instruction needed. Moreover, many teachers have had no access to these few books. Again, until recently not many courses of study and teachers’ manuals for textbook series have been of much help. It is not strange, therefore, that though meaningful arithmetic is adopted in a given school system, not all members of the teaching staff are well equipped to teach it. As a result, their pupils, denied a full and intelligent treatment of arithmetic as a body of rational ideas and procedures, have been unable to bring to computation all the aid that could come through understanding.” 

Now as you might be able to tell from that last paragraph, this does not come from a recent article.  This comes from an article originally published in 1956!!. The article was republished in 2003 by the National Council of Teachers of Mathematics in the Teaching Children’s Mathematics Journal.

I wanted you to be able to get your eyes on this article during the free trial because it’s so important to see how this debate and wondering about ‘meaningful arithmetic’, or more fittingly meaningful mathematics, was and still is a concern.  

As I re-read this portion it left me wondering, how do we help ensure that our students are not showing us an “illusion of understanding?”  Especially during this time when students are not physically present with you. It is hard to ensure that your students are developing that meaningful mathematics.

Everyone’s teaching situation is different right now.  Some of you are able to hop online with all your students, while others are only able to send home packets of work for theirs.  So I can’t give you the answers for your particular situation. Instead I do have some free trainings about laying a solid foundation (not memorization) and helping students develop their mathematical fluency.

Now some of the things I discuss and mention in the Building Elementary Math Fluency video series may not be applicable to your current situation and how you are able or unable to work with your students.  But ALL of it is applicable to when you get students back in your classroom.

Fluency does involve being fast and accurate, but it’s not memorization.  It’s the often missing part of flexibility. Flexibility is the most important aspect in developing true math fluency.  The Building Elementary Math Fluency video series starts off with a video about how fast ≠ fluent. Then you get videos about building fluency with math facts, about the math activities we can do that help build fluency, what it looks and sounds like when students have math fluency, and so much more.

But the videos are up for a limited time, so if you are listening to this later it may not be available.

Also on that page I will link up William Brownell’s article and the free trial to NCTM’s membership.

Stay safe and stay mentally healthy.

These episodes are sponsored by the online trainings that I do for elementary educators.  I have online number sense courses that thousands of teachers have gone through. Number Sense 101 is for Prek-2nd grade educators and Number Sense 201 is for 3rd-5th grade educators.  These courses help you understand the foundation of number sense, how it impacts students’ ability to become fluent in mathematics, and how to help your students build their number sense.  Registration for the courses will be opening soon. Go to to learn more about each course.

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As you start off the school year, I want you to keep in mind what is really important as we're trying to teach mathematics to our students.