RYSK: Brownell's The Place of Meaning in the Teaching of Arithmetic

This is the first in a series of posts describing "Research You Should Know" (RYSK).

Prior to the emergence of mathematics education as its own field of research in the 1960s and 1970s, those who chose to study the learning and teaching of mathematics were generally either mathematicians (George Polya, for example) or educational psychologists. William A. Brownell was one of the latter, having earned his doctorate in educational psychology in 1926 at the University of Chicago. Brownell worked at a number of universities but is best known for his work at Duke University from 1930 to 1949 and eleven years as dean of the School of Education at the University of California, Berkeley. Brownell died in May of 1977.

Brownell spent most of his career studying how children learn mathematics and authored a textbook series for the teaching of arithmetic. When the NCTM asked experts for nominations for articles to be included in their Classics in Mathematics Education Research, nine different articles were nominated. As the project leaders explained, "There was never a question whether to include an article by Brownell; the only issue to which of the many classic articles that he wrote to include" (Carpenter, Dossey, & Koehler, 2004).

In this article, Brownell defines the terms meaning of a thing and the meaning of a thing for something else. Or, in short, meaning of and meaning for. Using an example of the times, Brownell says that while he knows little about the workings of an atomic bomb (meaning of), he knows plenty about what having and using atomic bombs means for other things, like our culture (meaning for). In arithmetic, Brownell sees meanings of to be mathematical understandings, usually developed in classroom settings; meanings for are the connections to arithmetic we generally experience in every-day life outside of the classroom. Brownell insists that the distinction between the two is clear, although each type of meaning is relative, with varying degrees of complexity and depth.

Brownell saw, in 1947, too little teaching of arithmetic understanding. Common practice looked like this (you need only watch 20 seconds or so):

Through repetition, many students learned arithmetic. Many didn't. Again, learning was relative, not absolute, and Brownell never claimed that any students received zero understanding. What Brownell wanted was for students to see more meaning, which he organized into four categories:
  1. Meanings of whole numbers, fractions, decimals, percent, ratio, and proportion.
  2. Meanings of operations, including when to add, subtract, multiply, and divide.
  3. Meanings of principles and relationships of arithmetic, like the zero property, communitive property, etc.
  4. Meanings of place value that go beyond procedural "borrowing" and "carrying."
Brownell cited three sources for lack of meaning: (a) anecdotal evidence from adults who were poor at arithmetic, (b) poor military test results in arithmetic, and (c) the experience of secondary math teachers who receive students without arithmetical understanding. Brownell reasoned:
School personnel and, to some extent, the public at large are beginning to awaken to the fallacy of treating arithmetic as a tool subject. To classify arithmetic as a tool subject, or as a skill subject, or as a drill subject is to court disaster. Such characterizations virtually set mechanical skills and isolated facts as the major learning outcomes, prescribe drill as the method of teaching, and encourage memorization through repetitive practice as the chief or sole learning process.
Sound familiar? Of course. This kind of message has been echoed in NCTM materials at least since the Agenda for Action (1980) and common thinking can be found in articles such as Keith Devlin's Why We Should Reduce Skills Teaching in the Math Class.

So why didn't we (and don't we) teach arithmetic for meaning? Brownell anticipated four objections:
  1. Is it really necessary to understand meanings to learn arithmetic?
  2. Are these meanings too difficult for children to learn?
  3. Does it take too much time to teach meanings?
  4. Does learning meanings of and for arithmetic really improve ability?
To this, Brownell begins by asking why we teach arithmetic at all. If it is simply for students to be able to compute quickly and correctly, and nothing else, then the objections stand. However, if you believe that one of the goals of teaching arithmetic is to introduce to students a logical system of thinking, then meaning is necessary. Without it, computational efficiency -- which we should all still want -- will deteriorate once the drills are over. In this early day of research, Brownell apologizes for not citing "an impressively large body of competent research," but claims a strong case for meanings can still be made without it. Now, more than 60 years later, the research exists and the teaching of arithmetic looks very different. Students receive much better instruction about place value and use manipulatives like base-10 blocks, five-frames, and rekenreks to help their understanding.


Brownell, W. A. (1947). The place of meaning in the teaching of arithmetic. Elementary School Journal, 47, 256-265.

Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (2004). Introduction. In T. P. Carpenter, J. A. Dossey, & J. L. Koehler (Eds.), Classics in Mathematics Education Research (pp. 1-6). Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics [NCTM]. (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, VA. Retrieved from http://www.nctm.org/standards/content.aspx?id=17278.


  1. Brownell was co-author of a series called "Arithmetic We Need" which were written in the 50's. My school used them and I learned arithmetic through the series. I have the books; the explanations are similar to those found in Singapore Math. Interesting that Brownell is lauded for his "reform" ideas, and yet the 50's and 60's when his books were used is portrayed as an era similar to what is depicted in the video. Traditional math has been mischaracterized for years. And in 1947, math books were around that embodied the reform ideas that Brownell was espousing -- some written in the 30's. I have some of them.

    Ironically, although the teaching techniques depicted in the video are poor, the students who learned arithmetic in that era were likely able to perform computations that students entering college cannot perform without a calculator. And I'm talking about simple multiplication, and operations with fractions--this despite all the supposed advances of "reform" math with its spiral approach, student-centered, problem-based approach.

    You might be interested in this article:


  2. There is linkage between mastery of skills and understanding, especially when you get past the simple ideas in the video. I taught college algebra for years. Nobody can pass math with just rote knowledge. Mastery of skills is required in the "meaning of" domain.

    The modern math issue is more about top-down versus bottom-up approaches. There is no magic that can turn understandings into mastery of skills. Proper understanding only comes from the bottom up, not from a top-down focus on real world ("meaning for") examples. Math is not just some general way of thinking.

    In the 50's and 60's, I was able to get to calculus in high school with no help at home. This is virtually impossible now. My son will take AP calculus next year ONLY because of my help at home over the years. Go past place value and pie chart understandings of fractions to the mathematical identity level of understanding. Look at what understandings are required for a full and flexible knowledge of algebra. How does a pie chart understanding of fractions translate into the algebraic understandings of manipulating rational expressions. At this level, skills drive understanding, not the other way around.

    Keith Devlin says:

    "The paradox in this state of affairs is that Fact 2 probably provides the key to overcoming the obstacles stated in Fact 1. By providing our students with a good overall sense of mathematics, including the many major roles it plays in all our lives, we might well be able to provide the motivation the students need to spend some time acquiring basic skills."

    This is a pragmatic consideration and it clearly hasn't worked. It blames the student. It claims that if only students were motivated or engaged, then the problem would be solved. The problem with this is that this sort of top-down approach rarely gets the skills portion of the balance done. Parents have to finish the job at home. Everyday Math tells teachers to "trust the spiral" and just keep moving along. In my son's school this caused bright kids to get to fifth grade without mastering the times table. Some were still adding 7+8 on their fingers. I wouldn't have believed it unless I saw in myself. By seventh grade, their STEM career doors are virtually slammed shut. Still, educators talk about how these kids just need motivation and engagement.

    Another problem is that understandings are defined only at the most simple level. Students on the calculus (STEM) track in high school can't possibly succeed without understanding, and they sit in direct instruction classes and have to do large homework sets that work on both skills and understanding.

    Try sending out questionnaires to all parents of kids taking 8th grade algebra. Ask them exactly what they do at home. Our lower schools even had the temerity of sending home notes telling us parents to practice "math facts" with our kids while they were failing to drive understanding down to the most simple skills at school or with homework.