By Thomas Ultican 2/24/2017

It was wonderful that day I met Larry Lawrence at a Chicago Hotel frequented by Al Capone (The Drake Hotel). We were in Chicago for the National Public Education conference. I soon discovered two things: Larry only lives thirty miles up the beach from me in San Diego, County and he knows a lot about education. Larry participated in some of the key developments in the history of education methodology. Saturday, we met for lunch and I am still over-stimulated.

This is the third time we have met at the Ki Restaurant in Cardiff overlooking the Pacific Ocean. All three times, Larry has come prepared with notes including an informal agenda. This time, I was pleased that he wanted to begin by talking about a proposed fundamental reorganization of school which decentralizes power and democratizes operations. I had made such a proposal in my latest article which Larry had read. This fit well with his thinking that was influenced among other things by his time at UCLA’s lab school working with John Goodlad and Madelyn Hunter.

**The Math Wars**

Dr. Lawrence’s professional experience began with “new math.” 1956 was Larry’s third year at Occidental. He had finished the advance Calculus course and decided to register for a class called “Modern Algebra.” After his first day in class, he started studying the class materials and came across a concept he had never seen before, “one to one correspondence”. That concept is now considered an essential understanding for preschool aged children but in 1956 he searched fruitlessly throughout his dorm for anyone who knew what it meant.

Larry spoke about the experience,

“This illustrates the absolute mechanical nature of my math education to that point. This is something I have carried with me throughout my teaching career. How even the concepts that we might consider simple, may have no meaning for our students when they have no context for understanding.”

In 1958, Larry moved on to Teacher College, Columbia University to study math education under the tutelage of Professor Howard Fehr. An obituary in the New York Times said of Professor Fehr:

“Dr. Howard F. Fehr, professor emeritus of mathematics education at Columbia University Teachers College and a founder of new math in the 1960’s, died yesterday at his home in Manhattan after a long illness. He was 80 years old.

“Dr. Fehr, who retired from teaching in 1967 but continued in educational work, was a prolific author and an internationally known educator whose textbooks were used around the world. As the principal author in 1961 of a 246-page report titled ‘New Thinking in School Mathematics,’ Dr. Fehr helped introduce into American classrooms a concept of study and teaching that departed radically from traditional methods.”

Larry recalls Dr. Fehr’s class, “His ‘basic’ course laid out the fundamentals of the ‘new math’ – number systems, characteristics of a field, relations, functions, etc.”

After Teachers College (1959), Larry returned to his old high school, Morningside High in Inglewood, where he created one of the first high school calculus courses in California. In the summers of 1962 and 63, he attended a six weeks training course at the University of Illinois which was a program on how to use the math materials developed by Max Beberman and Herbert Vaugh.

Ralph A. Raimi states that “Max Beberman is generally regarded as the father of the New Math, his teaching and his curriculum project having achieved nationwide fame long before the 1957 Sputnik raised mathematics education to the level of a national priority.” Raimi also reports, “His thesis director at Columbia was Howard Fehr, famous then and later as an authority on the teaching of school mathematics, and a man who directed the PhD theses of many future professors of mathematics education.”

One of the problems for “new math” was it was often rushed into schools before materials were properly vetted or teachers were properly trained. The Stanford Mathematical Study Group (SMSG) under the direction of Edward G. Begle started producing curricular materials in 1958. Unfortunately this SMSG material became derided as “some math some garbage.”

“New math” also gets conflated with the progressive pedagogy. According to E. D. Hirsch, William Heard Kilpatrick was “the most influential introducer of progressive ideas into American schools of education.” (__The Schools We Need: Why We Don’t Have Them__, Double Day, 1996)

David R. Klein wrote A Brief History of American K-12 Mathematics Education in the 20th Century for Math Cognition. In it he wrote of Kilpatrick’s contribution to the math wars,

“Reflecting mainstream views of progressive education, Kilpatrick rejected the notion that the study of mathematics contributed to mental discipline. His view was that subjects should be taught to students based on their direct practical value, or if students independently wanted to learn those subjects. This point of view toward education comported well with the pedagogical methods endorsed by progressive education. Limiting education primarily to utilitarian skills sharply limited academic content, and this helped to justify the slow pace of student centered, discovery learning; the centerpiece of progressivism. Kilpatrick proposed that the study of algebra and geometry in high school be discontinued ‘except as an intellectual luxury.’”

Klein added,

“Meanwhile in 1920, the National Council of Teachers of Mathematics (NCTM) was founded, largely at the instigation of the MAA [Math Association of America]. The first NCTM president, C. M. Austin, made it clear that the organization would “keep the values and interests of mathematics before the educational world” and he urged that ‘curriculum studies and reforms and adjustments come from the teachers of mathematics rather than from the educational reformers.’”

The math wars were thus engaged in the early twentieth century. By the dawn of the 21^{st} century it appeared that the NCTM ideology had won the battle. Teaching math became based on teaching a set of discrete skills. However, today, much of the Common Core math teaching philosophy appears aligned with the progressive ideas of Dewey and Kilpatrick. Common Core also embraces the principles espoused by the proponents of “new math.” The “new math” was not really aligned with either side in the math wars but was more about teaching a cognitive understanding or foundation for learning mathematics and developing teaching methods.

Concerning the “new math,” Larry notes, “While it was a struggle for most teachers in the early years, the concepts have become part of the math curriculum of today.”

The influences on Professor Lawrence (Goodlad, Hunter, Fehr, Beberman, etc.) were experienced classroom teachers, developers of pedagogy and leaders in university teachers’ education departments. They were all exactly the kind of people that founders of the no-excuses charter school chains like John King, Doug Lemov, Mike Feinberg, Dave Levin and others disregarded. Instead, they turned to the economist Erik Hanushek for their guidance on good pedagogy.

**Organizing Schools**

In my article “Education Reform Musing” I proposed a democratized approach to school organization. Instead of a centralizing power in a principle, I advocated elevating the position of department head to lead circular development and establishing committees comprised of administrators, teachers, parents and students to set policies and resolve disputes. Larry was intrigued by this idea and wanted to discuss how it might fit into the structure that John Goodlad had introduced.

In 1959, the year before he became director of the lab school at UCLA, Goodlad wrote __The Non-graded Elementary School__. Amy Diniz of the University of Toronto comments:

“In the Non Graded Elementary School, Goodlad argued that the rigid graded education system is not designed to accommodate the realities of child development, including children’s abilities to develop skills at different rates to different levels. (Goodlad, 1963) Goodlad suggests that one limiting assumption embedded in the graded school structure is that children’s achievement patterns in different areas of study are going to be the same. However, in reality, most children progress quickly in certain subject areas while struggling in others. As Goodlad’s research demonstrates, it is very common to have a child in grade two have literacy skills of a grade three but math skills of a grade two. With a graded structure that assumes that a child will progress through each area of study at the same pace, a child is given no freedom to develop at the pace that is optimal for his/her needs.

“A second assumption in a graded system is that all students will learn the necessary skills within a year and then be ready to progress to the next predetermined level. In graded systems, students are all placed on an identical learning cycle with no room for diversity of learning patterns. (Kidd, 1973) Goodlad recognizes that under the graded system, the only options teachers have to adjust for students’ different learning capacities and rates are to either promote or hold back a student. However, there is ample evidence to suggest that both early promotion and non-promotion of a student are not strategies conducive for learning and development (Goodlad, 1963).”

I have taught remedial algebra at the high school level and have personally observed students learning math skills and concepts. Unfortunately, they were not learning fast enough to meet the state imposed standards, so, I was forced to give them failing grades. Worse than the graded system is the standardized system. Instead of meeting students where they are, we harm them because the standards do not match their cognitive development. School in America has long been a sorting system that separates the winners from the losers based on a meritocracy with elements of classism and racism coloring the decisions. Even if it were not flawed, the false perception that students achieve the same mental development at the same age convinces many students that they are not as valuable as others.

At lunch Professor Lawrence sketched out a possible alternative. Instead of age 5 kindergarten, age 6 first grade, age 7 second grade and so on, he postulated the possibility of leaning cohorts.

Cohort 1 for ages 5 to 8

Cohort 2 for ages 7 to 10

Cohort 3 for ages 9 to 12

Cohort 4 for ages 11 to 14

Cohort 5 for ages 13 to 16

Cohort 6 for ages 15 to 18

The overlapping age grouping is on purpose to allow teachers flexibility in moving students to new cohorts. The Diniz article describes Goodlad’s vision for this new structure:

“Two elements of Goodlad’s non-graded system include a longitudinal concept of curriculum and planned flexibility in grouping. Firstly, curriculum is centered on continual and sequential learning, with behavior and content running vertically through the curriculum (Goodlad, 1963). Longitudinal learning emphasizes that all skills learned are in fact base components of more complex skills to be learned in the future (Goodlad, 1963). Secondly, grouping is organized around achievement groups, interest groups, work-study groups or a combination of the three with some groupings being heterogeneous in skills (social sciences) and other groups being homogeneous in skill levels (reading).”

Developing a practical method for implementing Goodlab’s ideas was professor Lawrence’s job when in 1966 he joined the UCLA lab school which came under the purview of the UCLA Graduate School of Education. Lawrence says, “My task was to work within my team of teachers to develop a math program that would address the needs of our multiage, team-teaching organization. For the next few years, I explored a wide variety of programs that included SMSG materials and several others that began to be published in the late 60’s and early 70’s.”

John Goodlab was at the University of California, Los Angeles (UCLA) from 1960-1983, where he served as director of the Laboratory School and as dean of the Graduate School of Education (ranked first in America the last seven years of his tenure).

It is unfortunate that education reform became the domain of unqualified billionaires with no pedagogical understanding. It is time to take back our public school system. It is time to reengage with professionals. Privatizing public education is related to greed and foolishness. No excuse charters are related to abuse, segregation and arrogance. People who reject professionals for their own uninformed views are a menace to society. Let us build on the work of professionals like John Goodlab, Madilyn Hunter, Howard Fehr, Max Beberman and Larry Lawrence.

And, may I add, Maria Montessori. Her ideas were disparaged by Kilpatrick, perhaps because he wanted all the focus to be on his mentor, John Dewey, perhaps because she was a woman, a Catholic, and a foreigner, instead of a man, Protestant, and American. She wasn’t as focused on utilitarianism as Kilpatrick, or on the benefit for society, but only on the needs and interests of the child.

I quite agree that grades and groupings by grade level are deleterious, and that the teaching profession was very exciting when I was in college in the early 70’s and we were taught the concepts of Jerome Bruner and learned that education should be student-centered. All that went out the window starting in 1983 with A Nation at Risk, and, as you say, now all education policy is determined by billionaires – – or businessmen, former reporters, or economists like Hanushek – – playing amateur hour. No one listens to the only qualified professional experts: licensed practicing teachers.

Like many people, math was always my worst subject. To me that means something has to be very wrong with the way it’s taught. I started having trouble in seventh grade with rate, time and distance problems, It seemed so artificial and I didn’t see the point. I had algebra in eighth grade and loved solving for X, it was like a fun puzzle, though I never understood factoring polynomials. There seemed to be no formula, but all just trial and error, so I didn’t see the point, and even if there were a formula, I didn’t see the point. Geometry seemed to be a hodge-podge of totally unconnected things, none of which I understood what they were used for. We started with proofs, which the only way I could work them was to try to work backwards, which made no sense to me at all, and it seemed to have nothing to do with anything else in the course. It wasn’t until many years later that I learned that (maybe until the 20th century), all philosophers were geometricians and used these types of proofs to prove their points, like Spinoza using them to try to prove God exists. That was interesting. I loved congruent triangles, such a cool concept, but I never understood what sine and cosine were, and I only understood tangent metaphorically. All I remember about pre-calc was plugging numbers into formulas that had no meaning to me. I’ve always thought I would only understand math if I did some kind of History of Mathematics course, and went through all the steps that were done through history to get to where we are. I didn’t understand the point of quadratic equations; what I wanted to know was how somebody ever worked out the formula to begin with.

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