Barry Garelick - November 7, 2007
Columnist EducationNews.org
Part III of III

The Decline of the Test Scores and my own Private Cuba

Just as there are many factors that contribute to the rise in test scores, there are also many contributing to the decline that started in the mid-60's—a decline that included not only Iowa test scores, but SAT scores as well.By mid-1975 the decline became headline news in the New York Times bringing about an examination by various experts, and government agencies to find out the reasons.

Before addressing the general factors affecting all subject areas, some observations about the decline in math scores are worth examining. These observations stem from studies that came out prior to NCTM release in 1989 of its famous standards and well before the reliance on the National Assessment for Educational Progress (NAEP) exams as a bellwether for the achievement of students in the U.S.Just as Cuba is an example of a place that has been uninfluenced by American culture, the articles written prior to the NCTM standards are similarly untarnished and unaffected by the highly influential philosophy and vision represented by such standards. Similarly, NAEP scores were not yet elevated to the "gold standard" of testing that it enjoys today—Iowa test and SAT scores still ruled the roost.The math questions in the NAEP have been criticized for being extremely low level, and dumbed down, thus contributing perhaps to a false increase in math test scores in the past two decades. (Loveless, 2004)My focus on yesterday's papers has therefore allowed me to reside in my own private Cuba.I have glimpsed what may be one of the few relatively unbiased assessments of what was going on in math education, prior to "traditional math" having its current negative reputation.

At first glance, it is hard to ignore the fact that the decline of math scores from the mid-60's into the early 70's is remarkably coincident with the introduction of the New Math into curricula and schools, (early 60's to early 70's).The New Math brought an overly formal approach to math in the lower grades which included aspects of set theory.While the high school texts that emerged from that era were definitely better than what had come before, the lower grades suffered and math computation and procedural skills declined.

The continued decline in the 70's is remarkably coincident with the follow-up to New Math after its demise: books containing drills, basic skills and procedures in a thrown together hodge-podge of exercises that frequently lacked the central theory and motivation that the books in the 40's and 50's embodied.

The climb of math test scores in the 80's is remarkably coincident with the publication of "A Nation at Risk", the bad news report on the nation's schools by the National Commission on Excellence in Education and the "Back to Basics" movement that came about shortly thereafter, in which textbooks now started showing more attention to problem solving.Finally, the drop in computational skills starting in the 90's and into the 2000's is remarkably coincident with the release of NCTM's standards in 1989.

Nibbelink, et al (1986) examines the textbooks in use in each of the various eras.Nibbelink compares the numbers of problems of each such era, which he labels "Old Math" (40's and 50's), "New Math", (60's) "Aftermath" (early 70's) and "New Old Math" (mid 70's to 80's). The last era has also come to be known as the "Back to Basics" movement, which included more than just math.In general, in both the New Math and Aftermath eras, there were far fewer word problems than in the Old Math era.In the New Old Math era, the word problems increased to the level that were in texts from the Old Math era.Nibbelink offers a prescient warning about the texts of the "Back to Basics" era which was when his article was written, noting an unprecedented number of books force-feeding students "tightly fixed, multiple-step procedures for solving problems, sometimes of questionable fit" and draws a comparison to the centipede who in becoming too self-aware of how it walked, was no longer able to do so.Apropos to such comment, Loveless (2002) examines the Iowa test scores in the 80's and 90's, pointing out that although the non-computation (i.e,. problem solving) scores increased into the 90's, computation scores went into a swan dive starting in the 90's and into the 2000's, erasing the gains of the 80's and hitting low levels that hadn't been seen in more than two decades. 

Other Factors Influencing Test Scores and the Reinvention of John Dewy

During the era of test score decline, many social issues were emerging including increased drug use in the mid-60's, permissiveness, increase in divorces and single family homes, and changes in the demographics of schools.Other changes also went hand in hand with the radical ethic that emerged during the 60's.In particular, radical critics of schools such as Jonathan Kozol, brought accusations of sadistic and racist teachers, said to be hostile to children and who lacked innovation in pedagogy."Traditional" schooling was seen as an instrument of oppression and schools were recast in a new, "hipper" interpretation of what progressivism was supposed to be about.

With the revolution in education underway led by an unwavering belief that John Dewey "would have wanted it this way", a new student-centered philosophy prevailed bringing with it a lowering of standards, less homework assigned, classes such as Film Making and Cooking for Singles being offered, and requirements for English and History courses being dropped.High schools had long been divided into three tracks: academic or college-prep, vocational, and general.But with the requirements for graduation being diminished in the general track as a result of the student-centered fad, the general track saw an increase in students from 12 percent in the late 60's to 42 percent by the late 70's. (Ravitch, 2003).

The Back to Basics movement that began in the early 80's sought to reverse some of the effects of the fads and extremes of education and to some extent, this did in fact happen.The ideas of the progressives did not go away, however.Instead they continued to grow and flourish within education schools."Learning how to learn" continued to trump the learning of facts and content of an ever-developing world in which facts become obsolete.The mantras of student-centered learning continued as did inquiry-based learning and constructivism bringing with it the belief that "textbooks are a resource, not a curriculum".The effects of these education school-spawned beliefs have been noticeable, and not just in mathematics.In the last two decades, parents have become increasingly concerned and frustrated at the model building- and poster-dominated curricula of elementary, middle and high schools.Desks in a row are looked down upon, and discussions in small groups are now preferred.Students are encouraged to research and construct their own knowledge while teachers facilitate rather than teach. Such theory put into practice frequently amounts to an omission of instruction.Assessing students can be done on an informal basis by observing students solving problems in "real time" as opposed to answering a question on a test.Such approach is called "authentic assessment".The flip side, however, is that students are not given the instruction they need, but held accountable on traditional tests.Students (and parents) complain that the questions on tests includes material not covered by the teacher in class.

The burden on parents has therefore been increasing over the years, since they have often been forced to teach their children what isn't being taught in class, in order to help with homework.Interdisciplinary classes now also have become standard practice in many schools—aspects of history are taught in English classes, and essays are required in math classes.

Since the radicalization of the education schools since the mid-60's, what John Dewey and his protégés had sought to bring about gained more traction.When Brownell (1944) stated that "our attention as teachers is directed away from the processes by which children learn, while we are over-concerned about the product of learning", the result in the 1940's was an explanation of the process in tandem with practice exercises to ensure mastery.But the reinvented John Deweys now gave it a new interpretation: "Process is more important than product".The top-down, wholistic approach that emerged, placed the emphasis on big concepts at the expense of learning and mastering the basics needed to achieve the "critical thinking" and "higher order thinking skills" that they thought "process over product" would yield.The education school influence held sway in organizations such as NCTM during creation of the standards in 1989.The rest is history.

While bad teaching was incidental to the traditional method in earlier days, it has now become an inherent part of how much math is taught today. The NSF-sponsored texts with their emphasis on multiple algorithms, open-ended questions, top-down discovery and spiral learning force teachers to omit explicit instruction, although more experienced and knowledgable teachers may supplement (or supplant) such texts.

Even when so-called "traditional" texts are used, there are problems.The vision of NCTM embodied in its standards have permeated many standard textbooks published today.Books now tend to be written with fewer problems, contain rambly and often incoherent sequences of topics, and a de-emphasis on basic skills and procedures.Finally, the most insidious aspect of the problems with mat teach is that the reform approach to teaching math is being taught in education schools, thus effecting a "work-around" to those few math textbooks that actually have merit.

A case in point is the issue of "real world" problems. When talking about higher level math courses such as algebra and beyond, the criticism towards textbooks that reformists levy is that word problems are often not relevant to real world situations. A book used in education schools in courses for future teachers of high school math is "Teaching Mathematics in Secondary and Middle School" (Cangelosi, 2003).In it, Dr. Cangelosi guides future teachers away from the perils of word problems that are contained in algebra texts.He raises a flag about textbook word problems:"To solve a real-life problem, students must clarify the questions posed by problems, often in terms of more specific questions about quantities.With textbook word problems, the specific questions involving quantities … are typically articulated for students.Thus, instead of taxing students' abilities to formulate questions, what is taxed is the students' reading comprehension skill."

Cangelosi's solution is to include problems in which students collect their own data to draw conclusions about a particular problem. For example, students might collect measurements of arm span and heights of various students and plot the relationships (using a graphing calculator).The problem would then be to define a formula that best describes the relationship, and with facilitation of small groups in class, decide whether it is linear.Such types of exercises then turn algebra into a statistical approach, at the expense of problems that more directly address the principles of algebra.Such principleswill later generalize to solve a variety of problem types that crop up in science and engineering.As it is, however, "real world" problems such as those exhorted by Cangelosi and others leave students with the impression that math is an empirical science and its precision is out of place in a messy world that cannot be defined by mathematics.

Future math teachers are thus indoctrinated in education schools so they can deal with traditional text books, such as algebra books by Mary Dolciani which are quite rigorous, and are still in use in some schools. If a teacher is "stuck" with Dolciani, Dr. Cangelosi and others have now given teachers work-arounds so they can effect a hands-on, learning environment.

Fortunately, excellent math teachers continue to exist who know enough to ignore what they are taught in education school. Unfortunately, there are also others who believe in what they've learned, falling prey to what is called a "balanced approach" to teaching math.This generally refers to relying on a direct approach and drill of algorithmic skills on some days, but a discovery approach to learning concepts on others, with real world problems thrown in the mix. "Meaningful learning" no longer means what it did when Brownell wrote about it.

No Golden Age but Perhaps a Silver Lining

The increase in math test scores reported in the Iowa tests from the 40's through mid 60's do reveal differences in the approaches taken in textbooks and in the teaching itself.This is not to say that whatever worked back then is the key; there were aspects to both teaching and texts that could definitely be improved.But why not look at what worked in the past, as well as what is working in countries such as Singapore and Japan rather than dismissing these as "traditional approaches" which don't work?The traditional approaches of the past have worked; that they can work better is not a reason for absolute rejection.

The mischaracterization of traditional math fails to capture the nuance of instruction and scaffolding that many teachers (including mine) engaged in.They didn't just stand there and lecture; they asked us questions and goaded us into thinking, providing us with prompts and support through many worked examples.This is a form of direct instruction, and it was combined with cumulative reviews and mastery learning.It is interesting therefore to learn that direct instruction and mastery learning are recommended methods of teaching for students with learning disabilities. (Rosenberg, et al., 2008)It is also interesting to note that over the past two decades, the number of students with learning disabilities has increased.In 2006, approximately 2.6 million students were identified with learning disabilities, more than three times as manyas were identified in 1976-1977. Although one reason for this growth might be better means of diagnoses of specific disorders, there has still been growth.Between 1990 and 2004, 650,000 additional students were identified with learning disabilities, representing a 31% increase at a time when the overall student population grew by only 15%.(U.S. Department of Education, 2006).

The increase in the number of students with learning disabilities raises the interesting question (if not uncomfortable for some), of whether the older way of teaching (direct instruction and mastery learning) may have had unintended benefits. According to Rosenberg, et. al. (2008), one factor associated with the identification of students with learning disabilities is the lack of access to effective instruction.Rosenberg et. al, also note that up to 50% of students with learning disabilities have been shown to overcome their learning difficulties when given high-quality instruction. Is the shift toward inquiry-based teaching resulting in more students being identified with learning disabilities?Are these students who in earlier days would have swum with the rest of the pack?

To answer this question would take a good bit of solid research.I hold it out as a research project for anyone willing to take it on, perhaps as a dissertation for a PhD.It would certainly provide some research that Sherry Fraser could cite.At the very least it might even result in helping people learn.

References

Aiken, D. J., Henderson, K.B., & Pingry, R.E. (1957).Algebra: Its Big Ideas and Basic Skills.New York: McGraw-Hill Book Co.

Brownell, W. A. (1944).The Progressive Nature of Learning in Mathematics.Mathematics Teacher. Vol. 37.National Council of Teachers of Mathematics. April

Brownell, W. A., Buswell, G.T., Sauble, I. (1955a) Teaching Arithmetic We Need. Boston: Ginn and Company

Brownell, W.A., Buswell, G.T., Sauble, I. (1955b) Arithmetic We Need (Grade 6). Boston: Ginn and Company

Brueckner, L. J., Anderson, C. J., Banting, G. O., Merton, E. L.(1941).The New Curriculum Arithmetics (Grade Seven).Topeka: The State of Kansas; W. C. Austin, State Printer

Cangelosi, J.S. 2003. Teaching Mathematics in Secondary and Middle School: An Interactive Approach (3rd Edition).Columbus: Pearson, Merill Prentice Hall

Clark, C.W. ,Moser, H.E. (1952) Growth in Arithmetic (Grade 3).World Book Co.

Congressional Budget Office. 1986.Trends in Educational Achievement. Prepared by Daniel Koretz of Congressional Budget Office's Human Resources and Community Development Division.Congress of the United States.Available at: http://www.cbo.gov/ftpdocs/59xx/doc5965/doc11b-Entire.pdf

Fraser, S. (2006).National Math Panel Testimony.Stanford, California. November 6.Available at: http://www.ed.gov/about/bdscomm/list/mathpanel/4th-meeting/presentations/sherry-fraser.pdf

Hernishfeger, A. and Wiley, D.E (1976).Achievement Test Scores Drop. So What? Educational Researcher, Vol. 5, No. 3. March.

Hoover, H.D, Dunbar, S. B., Frisbie, D.A, Oberley, K.R., Ordman, V. L. Naylor, R. J., Bray, G. B., Lewis, J.C., Qualls, A.L., Mengeling, M.A., and Shannon, G.P.2003.The Iowa Tests: Guide to Research and Development; Prepared at University of Iowa; Riverdale Publishing; pp. 55-60.

Loveless, T. (2002)The 2002 Brown Center Report on American Education.Brown Center on Education Policy.The Brookings Institution.September.

Loveless, T. (2004)The 2004 Brown Center Report on American Education.Brown Center on Education Policy.The Brookings Institution.September.

Mayfield, K. H., Chase, P. N. (2002).The Effects of Cumulative Practice on Mathematics Problem Solving.Journal of Applied Behavior Analysis.Vol. 35, No. 2.Summer.

Morton, R. L., Gray, M. Springstun, E., Schaff., W.L.(1946). Making Sure of Arithmetic (Grade Six).New York: Silver Burdett Co.

Nibbelink, W. H., Stockdale, S.R., Hoover, H.D., Mangru, M. (1986). Problem Solving in the Elementary Grades: Textbook Practices and Achievement Trends Over the Past Thirty Years. Arithmetic Teacher; Vol. 35, No. 1.September.

Ravitch, D.(2003) The Test of Time.Education Next. Spring.Available at: http://media.hoover.org/documents/ednext20032_32.pdf

Rosenberg, Michael S., Westling, D.L., McLeskey, J. 2008. Special Education for Today's Teachers: An Introduction. Columbus: Pearson, Merrill Prentice Hall.

Rothstein, Richard. 2000. Add Social Changes to the Factors Affecting Declining Test Scores. New York Times; October 25.

Rothstein, Richard. 1998.The Way We Were?: The Myths and Realities of America's Student Achievement (Century Foundation/Twentieth Century Fund Report) "Century Foundation.Brookings Press.

U.S. Department of Education. 2006.Individuals with Disabilities Eduation Act (IDEA) data.Available at: www.ideadata.org/index.html

Published November 7, 2007

It Works for Me: An Exploration of Traditional Math Part 1

It Works for Me: An Exploration of Traditional Math Part II