Barry Garelick - November 6, 2007
Columnist EducationNews.org 
Part II of III

The Rise of Test Scores

A number of factors may be at play to cause an overall increase in all subjects in general, and mathematics in particular.  Nibbelink, et. al (1986) suggests a few such as increased standards for teacher certification, better child health care, better snow removal equipment keeping rural roads to schools open, and a developing belief that mathematics and science was extremely important.  This last was due to a noticeable lack of mathematic and scientific literacy in draftees during World War II, which ultimately resulted in colleges and universities in the 50’s relaying to high schools what proficiencies in mathematics they expected from entering freshmen.   (Such “trickle down” curriculum expectation fed into a movement to improve algebra texts and classes that began in the early 50’s and was pioneered by a math teacher named Max Beberman.)  

While a complex mix of factors may be responsible for the general increase in math achievement scores, there is also evidence that there were changes going on with math textbooks during this period as well. The major players among math education reformers of the 20’s through the 50’s include Leo Brueckner, Robert Lee Morton, Foster Grossnickle, Arnold E. Moser, Guy T. Buswell and William A. Brownell.  Brownell, spoken well of by NCTM and various luminaries in today’s reform movement, was the key reformer of the early twentieth century and promoted what he called meaningful learning; i.e., teaching mathematics as a process, rather than a series of end products of isolated facts and procedures to be committed to memory.  If this sounds like what the reformers are talking about today, it is because the complaints levied against how mathematics is taught, like the complaints about education in general through the years, have been perennial.  What is often not mentioned when these complaints are “replayed” is that there have also been perennial solutions and some of these solutions have actually been effective. 

Brownell led the charge against the isolated, rote-memory type of math teaching that came about in large part through the books and efforts of E. L. Thorndike, another figure of education at that time.   The reformers listed above, including Brownell, all wrote math texts that were in use from the 30’s through the 60’s.  The later books were written by Brownell (with Guy T. Buswell and Irene Sauble) starting in the mid-50’s in a series called “Arithmetic We Need”.  I’m familiar with this series because they were the books I used when I was in school.  I have copies of these and other books in use by all the authors.  All the books give explanations of what is going on with specific mathematical procedures, and topics were presented in a logical sequence that allowed building upon previously learned and mastered material.  But what is particularly interesting are the explanations in the teacher’s manuals and prefaces to these books:

• From “Making Sure of Arithmetic” (Grade 6):  “Each new process is explained in the simplest terms, utilizing every graphic aid possible.  From the beginning, meaning and relationship are emphasized.  As a result the pupil gains not only skill but skill with understanding.” (Morton, et. al., 1946)
• From “The New Curriculum Arithmetics” (Grade 7) “A program of mixed and cumulative practice exercises insures mastery and retention of the processes and topics studied.” (Brueckner, et al, 1941)
• From “Growth in Arithmetic” (Grade 3):  A comparison chart in the teacher’s edition showing the difference between the older (Thorndike-derived) textbooks and this one: “Older: Taught as facts, skills, and habits of procedure; Newer: Taught to emphasize meanings, principles, and relationships. Facts and skills developed after understanding.” (Clark, et al, 1952)
• From “Teaching Arithmetic We Need” (Grade 5) “Each book in this series is built upon a conception of arithmetic that involves two aims, the social aim and the mathematical aim.  Adherence to the latter aim requires that children see sense in what they learn.”  (Brownell, et. al., 1955a)

While the above sounds like they may have come from the introductions of various reform texts, the main difference is that 1) there were actual textbooks that students could read (as opposed to workbooks with problems but little to no written explanations of what is going on) and 2) these textbooks contained actual content that required mastery by the student. 

You may well be wondering why, the books are so heavy on drill if these reformers came from the school of “progressivism” which embodied the notion of “teaching the child” and focusing on a student-centered curriculum.  The inclusion of drills in these books is not inconsistent with the philosophy of the progressives.  The reformers believed in teaching the child, but through “meaningful learning” which in Brownell’s and others’ view, meant teaching the concept with examples of how (and why) it worked, and then providing the student to practice the procedure to ensure both understanding and mastery of the skill involved with applying the concept.

The drills in the books from the 40’s, 50’s and early 60’s are varied, mixed, and cumulative (i.e., continually including problems that were addressed earlier within new material).  That they are cumulative is important, given recent research showing the positive effects of such practice on problem solving. (Mayfield and Chase, 2002). Also, there are drills that simply ask students to identify the operations needed to solve the problems, to ensure that they know when to apply, say, multiplication as opposed to division in solving a problem.

I would therefore add to the list of possible factors influencing the upward trends in achievement scores from the 40’s through the mid-60’s the textbooks in use, and the implementation of the theories behind them.  This is not to say that the traditional math of such time was perfect.  If I had to compare the “Arithmetic We Need” texts that I used with Saxon Math, or the math program used in Singapore, I would say the latter two are superior with much more challenging word problems.   I can say, however, that the essentials of math were covered well, which would include place value, why a particular algorithm worked, thorough application of fractions and multiplication and division of fractions (similar to Singapore’s approach) and application of procedures to solving word problems.  Here are two problems from the sixth grade book of “Arithmetic We Need”:

• “How many glass containers holding 3/16 quarts can be filled with water from a quart bottle which is ¾ full?”
• “If it takes 1and  ½  hours to drive to the city,  what part of the distance will Bill and his father drive in ¾ hour?”

(Brownell, et.al., 1955b)

It cannot be denied that some teachers did not follow the texts and insisted on an approach that relied on  rote-memorization and math problems isolated from word problems—an approach that Thorndike promoted and against which Brownell and others rebelled.,  But neither Brownell, the other reformers of those times, nor mathematicians, asked the teachers to teach math that way. Any lack of continuity between textbooks used between grades was also not the fault of authors or mathematicians. If by "traditional method" and "old way of teaching math" people mean poor teaching and bad planning, it should be noted that such was incidental to and independent of the textbooks used and the philosophy put forth by the reformers.

It Works for Me: An Exploration of Traditional Math Part I
It Works for Me: An Exploration of Traditional Math Part III