The first in a series that every Parent, Student and Educator Can Read to Better Understand the Current American Math Problem and Concerns about “Flipped Classroom”.

I was recently sent a link to a comprehensive article about the history of mathematics in the United States. Despite being titled ‘A Brief History of American K-12 Mathematics Education in the 20th Century‘, the article is, proportionally, as long as its title. A Brief History of American K-12 Mathematics Education in the 20th Century It is worth a read in its entirety, but there are several points made in the article that dovetail with what I believe to be issues with the way Berkeley Heights school district currently approaches mathematics education.

Over the past two weeks I’ve had several conversations with both parents and students. Math was the hot topic in many of those conversations. When Dr. Kelly Curtiss held a parent information session at Columbia Middle School (Berkeley Heights, NJ) at the beginning of the year, attendance was so high that parents were standing at the back of the room. While there are generally only two or three residents attending any board of education meeting, and at typical PTO meetings only officers and event chairs show up, the discussions and participation around mathematics shows many in the district have continuing concerns. In addition to the program that was introduced at Dr. Curtiss’ information session – a program that appears very similar to Building the Thinking Classroom (BTC) – a new method of teaching has now entered the fray: Flipped Classroom.

Students whose teachers are using this method view a video at home, then work in groups in the classroom to solve problems. It’s not going well. On a positive note, it seems the use of BTC-like practices are no longer the main driver of mathematics education in many classrooms, rather they are being used as “one tool”, which is how the district told us they were meant to be handled. Taking a look at how mathematics has historically been taught may help to give our district a framework for how we can continue to improve.

In the opening paragraphs of A Brief History, the author notes,

Throughout the 20th century the “professional students of education” have militated for child centered discovery learning, and against systematic practice and teacher directed instruction. In some cases, progressivist math programs of the 1990s were intentionally without student textbooks, since books might interfere with student discovery. The essence of the dictum from educators of the 1990s and late 1980s, that the teacher should be “a guide on the side and not a sage on the stage,” was already captured in a statement from the principal of one of John Dewey’s “schools of tomorrow” from the 1920s:

The teacher’s arbitrary assignment of the next ten pages in history, or nine problems in arithmetic, or certain descriptions in geography, cannot be felt by the pupil as a real problem and a personal problem.”

It would be a mistake to think of the major conflicts in education as disagreements over the most effective ways to teach. Broadly speaking, the education wars of the past century are best understood as a protracted struggle between content and pedagogy. At first glance, such a dichotomy seems unthinkable. There should no more be conflict between content and pedagogy than between one’s right foot and left foot. They should work in tandem toward the same end, and avoid tripping each other. Content is the answer to the question of what to teach, while pedagogy answers the question of how to teach.

The trouble comes with the first step. Do we lead with the right foot or the left? If content decisions come first, then the choices of pedagogy may be limited. A choice of concentrated content precludes too much student centered, discovery learning, because that particular pedagogy requires more time than stiff content requirements would allow. In the same way, the choice of a pedagogy can naturally limit the amount of content that can be presented to students. Therein lies the source of the conflict.”

This appears to be where we are within our district, as the focus is currently on teaching method. The idea that students must learn to think critically through self-discovery sounds wonderful on paper. In reality, we are doing a disservice to them as they struggle to find a time balance in their lives. Three students in the Flipped Classroom model have reported frequently spending between one and two hours a night on math alone. On the subject of self- or group-discovery, one recent graduate told me, “I don’t understand the reluctance to just teach. We have the ability to stand on the shoulders of those who came before us, so why do we need to discover the Pythagorean Theorem on our own. What a waste of time.”

The results may be hard to track accurately in the short-term, but it appears we are not largely improving. In fact, Berkeley Heights has gained a bit of a reputation in the wider community for students’ lack of mathematical knowledge. An instructor at a test preparation company told a student, “Kids from Berkeley Heights can’t do math.”

Returning to the historical article, one of the most influential mathematical educators in the early 1900’s, William Heard Kilpatrick, argued for the progressive idea that mathematics need only be taught to students based on practical value of what they will use in their lives, and that algebra and geometry should be discontinued “except as an intellectual luxury.” This was supported by educators at the Teachers College of Columbia University. The Mathematical Association of America (MAA) responded and in 1920, supported by the MAA, the National Council of Teachers of Mathematics (NCTM) was formed. They argued for keeping “the values and interests of mathematics before the educational world” and for math curriculum to come from teachers, rather than educational reformers, yet it didn’t take long before the organization “attracted to its membership and to its leadership those in positions much more subject to the influence and pressure of the professional reform movements.” In the 1930’s the theme would be curriculum determined by the needs and interests of children, with the slogan “We teach children, not subject matter.”

In the 1940’s it became apparent that many graduates lacked even the skills necessary for basic bookkeeping. Amazingly, this led to an even more progressive ideology in which it was said 60% of the population lacked the capacity for college work and that students should be taught “home, shop, store, citizenship, and health.” By the end of that decade, advances in technology had highlighted the need for students to be taught mathematics. Critic Mortimer Smith wrote, in his 1949 book Madly They Teach:

…those who make up the staffs of the schools and colleges of education, and the administrators and teachers whom they train to run the system, have a truly amazing uniformity of opinion regarding the aims, the content, and the methods of education. They constitute a cohesive body of believers with a clearly formulated set of dogmas and doctrines, and they are perpetuating the faith by seeing to it through state laws and the rules of state departments of education, that only those teachers and administrators are certified who have been trained in the correct dogma.

New Math emerged in the 1950’s, but it wasn’t until the launch of USSR’s Sputnik that the US made serious inroads toward educating students in mathematics and science. The American Mathematical Society set up the School Mathematics Study Group (SMSG), and the NCTM created the Secondary School Curriculum Committee. This movement saw the introduction of Calculus in the high school curriculum, but at the same time, much of the subject matter taught was too abstract and teachers were not equipped to handle the material. This led to much public and institutional criticism, and by the 1970’s New Math was dead.

Taking the place of New Math was the Open Education Movement based on Summerhill, a progressive school in England where students determined what they would learn. The premise was “Whether a school has or has not a special method for teaching long division is of no significance, for long division is of no importance except to those who want to learn it. And the child who wants to learn long division will learn it no matter how it is taught.” This movement was particularly destructive for poor students whose parents had limited resources to give their children what they were missing with this non-curriculum.

“The National Council of Teachers of Mathematics released An Agenda for Action in 1980. They deemphasized work with pencil and paper and complete mastery of skills, with the assumption that calculators would remove this need. It was thought that needing skill mastery would interfere with learning problem-solving strategies. ”

The report also recommended that “Team efforts in problem solving should be common place in elementary school classrooms.” These directions would later become issues of contention in the math wars of the 1990s.” At the same time, An Agenda for Action was overshadowed by A Nation at Risk, written by a commission appointed by the US Secretary of Education. It addressed a wide variety of education issues, including specific shortcomings in mathematics education, noting that remedial math courses at 4-year universities had increased 72%. The report also stated “The teacher preparation curriculum is weighted heavily with courses in “educational methods” at the expense of courses in subjects to be taught” and noted a shortage of math and science teachers.

In 1989, NCTM took their agenda and established Curriculum and Evaluation Standards for School Mathematics. Much like the progressive ideals during the first two decades of the century, the standards emphasized self-discovery and deemphasized rote practice, rote memorization of rules, and “teaching by telling”. A new term, constructivism, emerged. E.D. Hirsch Jr. provided a useful definition in his book, The Schools We Need: Why We Don’t Have Them, which begins as follows:

“Constructivism” A psychological term used by educational specialists to sanction the practice of “self-paced learning” and “discovery learning.” The term implies that only constructed knowledge–knowledge which one finds out for one’s self–is truly integrated and understood. It is certainly true that such knowledge is very likely to be remembered and understood, but it is not the case, as constructivists imply, that only such self-discovered knowledge will be reliably understood and remembered. This incorrect claim plays on an ambiguity between the technical and nontechnical uses of the term “construct” in the psychological literature…

Criticisms, however, were not well-tolerated in the educational community and amongst academics.

At the same time the NCTM published their standards, a legislative call went out for the US to adopt national standards for education. The author writes “The NCTM Standards were immediately and perfunctorily endorsed by a long list of prominent organizations such as the American Mathematical Society, the Mathematical Association of America, and the Council of Scientific Society Presidents.” Subsequent documents from NCTM focused more narrowly on pedagogy and testing. The National Science Foundation (NSF) became instrumental in ensuring the standards were implemented, incentivizing states by giving grants to state agencies. The NSF then created the Urban Systemic Initiative, followed by the Rural Systemic Initiative, targeting education at local levels.

As the author writes,

The NSF was clear in its support of the NCTM Standards and of progressive education. Children should learn through group-based discovery with the help of manipulatives and calculators. Earlier research funded by the NSF, such as “Project Follow Through,” which reached very different conclusions about what works best in the classroom, would not be considered. Regardless of what cognitive psychology might say about teaching methodologies, only constructivist programs would be supported.

The 1990’s “Math Wars” were kicked off, in part, by parents in Princeton, NJ as they fought against the NCTM Standards. “In 1991 a group of about 250 parents of school children in Princeton, New Jersey petitioned the board of education for a more systematic and challenging math program. They found the one in use to be vague and weak. Many of the teachers did not even use textbooks. When parents asked about what was being taught in the classrooms, they were told that the curriculum was not very important, that “one size does not fit all,” and, repeating the dictum of 1930s Progressivists, that the teachers were there to “teach children, not curricula.”

When parents complained of deficiencies in what little curriculum even existed, they were treated as if their cases were new and unrelated to other complaints. These responses have been reported by parents in many other school districts as well.Test scores in Princeton were among the highest in the state, but that was not the result of a well designed academic program. Many highly educated parents, including Princeton University faculty, were providing tutoring and enrichment for their own children. Other children with limited resources in the Princeton Regional School system did not fare well in this highly progressivist environment.”

Despite several of these parents (one of them a theoretical physicist at the Institute for Advanced Study in Princeton) eventually taking over the Princeton Board of Education, they were unable to affect meaningful change, and they went on to establish the Princeton Charter School. Efforts of parents in California whose credentials in math and science could not be ignored resulted in some of the more effective challenges to the Standards. In 1985, California had implemented its own version of similar standards and with that set and adoption of the NCTM program, the results “pointed to a decrease in Stanford Achievement Test scores coinciding with the implementation of “whole math” in district schools. From 1992 to 1994 the average overall student score for 8th grade math students had decreased from the 91st national percentile rank to the 81st. The decrease was more dramatic on the portion of the exam that tested computation. On that portion the scores dropped from the 86th percentile in 1992 to the 58th percentile in 1994.”

Groups in California were somewhat successful at challenging this progressive path, convincing the state to rewrite standards for schools, yet the results were still fraught with issues. The article reports,

In January 1997, a committee called the Academic Content and Performance Standards Commission (Standards Commission) was charged with writing mathematics (and other subject matter) standards for California and submitting its draft to the State Board of Education for final approval. The committee consisted of non expert citizens appointed through a political process.

The majority of the Standards Commissioners were largely in agreement with the constructivist policies of the past. The result was a set of standards submitted to the Board in the Fall of 1997 that not only embraced the constructivist methods that California was trying to escape, but was also incoherent and full of mathematical errors.

The State Board then turned to several Stanford mathematicians for help, and the resulting framework was submitted and adopted in 1997. In 1998 the Fordham Foundation ran an analysis of mathematics standards from 46 states and also from Japan. California’s new standards outranked the entire field. However, the math war was far from over.

NCTM lashed out against the new standards, writing, “California’s state board of education unanimously approved curriculum standards that emphasize basic skills and de-emphasize creative problem solving, procedural skills, and critical thinking.” They called the material “yesterday’s content” and said the vision bore no relation to reality. Proponents of the new standards, including the chairs of mathematics at Stanford and Caltec and over 100 mathematics professors, hit back by adding their names to an open letter of support.

The article ends with the year 2000, the year NCTM revised its first set of standards and issued the Principles and Standards for School Mathematics. The author, at that time, expressed hope that efforts started in the 1990’s to bring mathematics education to more of a compromise between the constructivism model and the practice of basic skills and foundations in algebraic skills would continue to flourish. He writes,

In an era of international competition, it is unlikely that the public will tolerate such trends indefinitely. It was the broad implementation of the NCTM reforms themselves that created the resistance to them.

Ironically, the extraordinary success in disseminating progressivist mathematics programs may, in the long run, be the principal reason for the demise of progressivism in mathematics education.

Part two of this subject will focus again on California and track what has transpired from the turn of the century to today.

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