Liping Ma: elementary mathematics and core subject structure

 Woman teaching geometry, from Euclid's Elements.
Woman teaching geometry,
from Euclid's Elements.

(Photo credit: Wikipedia)
Years ago, I was listening to mathy homeschooling parents talk about Singapore Math and, like so many others, I was reading a book by Liping Ma, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United StatesThis book, and her others, have been hugely influential and embraced "across the spectrum" by those concerned with the teaching of mathematics. Dr. Ma clearly shows that most US teachers lack the grasp of elementary mathematics that most Chinese teachers have; US teachers lack procedural knowledge of basic algorithms, they cannot generate multiple representations of the concept, and few if any display deep insight into the structures of elementary mathematics, what Ma calls the profound understanding of mathematics (PUFM) that at least 10% of Chinese teacher do.

This was true in spite of the fact that the Chinese math teachers had much had less overall schooling, often 11 to 12 years total compared with US teachers with an average of 16 to 18 years of schooling including a full BA. Many teachers in Dr. Ma's book are barely high school graduates in terms of years spent in school; they began Normal school training as teachers after the 9th grade. These teachers in China, with fewer than twelve years of total schooling, working among themselves with concise texts, developed their knowledge of elementary arithmetic and its structure, some attaining over time a profound understanding of elementary arithmetic.

Dr. Ma believes Chinese teachers start with a stronger comprehension of elementary arithmetic before they begin, in the 10th grade, to study teaching but they also have substantial time and resources for practice, collaboration and study throughout their professional career. Dr. Ma documented, and many other sources have also confirmed, the deep preparation, lighter teaching load, and collaborative work of math teachers in Asia. There is no stronger case for allowing teachers to have the time and resources to practice, collaborate and study, daily and peer-to-peer than this book. There are lessons about how much schooling is really needed, too. We have essentially a credential industry based on the factory model and there is no structural push back against expansion of time and energy.

I should note that homeschoolers helped build the popularity of the Singapore math curriculum and it is now used by public schools in many states. This is another accomplishment of homeschooling often overlooked. Families are not too stupid to understand curriculum, indeed, they have proved to be strong supporters of academic quality, though school systems insist that families should leave complex stuff like this to professionals, who know better. (The work of Liping Ma suggests otherwise.)
from Liping Ma's website
The Structure of Elementary School Mathematics
Dr. Ma has continues her important work and her recent article,  A Critique of the Structure of U.S. Elementary School Mathematics, provides a deep critique of the structure of elementary mathematics in the US. It is a remarkable document. The contrasting models in Fig 1 convey the full weight of her critique: a three-dimensional figure of a cylinder versus the flat, two-dimensional figure of the dangling strands. The cylinder clearly shows the coherent body of knowledge that elementary mathematics maps in Chinese curricula. Chinese elementary mathematics provides an entire dimension that elementary math in the US lacks: a core subject structure, arithmetic, that is self-contained. Dr. Ma provides a diagram that shows the organization of school arithmetic in the primary grades, one through six, and she demonstrates the dominance of arithmetic within that organization. She notes the careful construction of the curriculum and its attention to the relationship of topics.

Dr. Ma then proposes that the essential feature of a core subject structure is an underlying system of definitions. One test of a strong definition system is the ability to do problems that require a complete set of definitions. A well-known problem from Singapore Math, that had many Americans oddly puzzled that they could not solve the problem with arithmetic but only with algebra, shows the lack of a complete set of definitions. This is the root of the problem with learning fractions in schools, a problem unknown in the popular press even in this recent article at the WSJ.

I saw these differences when using the Singapore math and New Elementary Mathematics textbooks. (These are the popular version of Singapore math and not the programs used in China. Significantly, Dr. Ma has to refer to curricula from before the 2001, when China moved toward its own version of the strand structure.) It is a very different experience to learn an algorithm and practice it and then do another one, versus learning an algorithm that is examined, tested, verified, extended and elaborated in a mathematical way. The difference is profound in impact. Problems are not practice sets but are instead the core work. These problems specifically work to build insight into mathematical structures and ways of thinking. The careful and finely-detailed construction is evident, challenges are routine, and the problems are rewarding to work. Elementary mathematics can still be mathematics and the depth of core content structure makes the subject far more interesting. A noticeable feature of this new model is the way it does not dumb down mathematics but finds ways to have children do mathematics instead of rote work.

In my experience, they are almost like two different subjects, in that the focus of what is being taught is so different. In a program with this core subject structure, even when learning a specific algorithm, you become aware of a larger set of relationships: it is a circumscribed body of knowledge and not a set of skills. This is what prepares the learner for higher mathematics: not only is there a strong foundation of arithmetic but there is also the experience of building a wider set of mathematical relationships that develop the ability to think abstractly.

It is important to grasp that what Dr. Ma's work shows is that US schools do not use arithmetic as the core subject structure of elementary mathematics. The US did have arithmetic in elementary education and that was eroded and then lost (as discussed below) but the US never had the fully developed model she discusses. It is a new model. More discussion of Dr. Ma's recent article:
New analysis of US elementary school mathematics finds half-century of problematic 'strands' structure | e! Science News: "In her article, Ma notes that, in many countries where students do well in mathematics, elementary mathematics has school arithmetic as its main organizing structure. School arithmetic is developed as a self-contained subject consisting of whole numbers and fractions, with the whole numbers forming the basis for understanding of fractions.  [...]
Examining developments in U.S. mathematics education going back to the 19th century, Ma notes that although U.S. scholars made significant contributions to school arithmetic, the U.S. never had, as some other countries do, a well-developed school arithmetic. Nevertheless, arithmetic was the core of elementary mathematics in the U.S. for almost one hundred years. Ma describes how this began to change during the 20th century with the advent of the "New Math" of the 1960s*** and the NCTM Standards of the 1990s. Among the effects of the strands structure are instability of curricular content, discontinuity in instruction, and incoherence in concepts.
"In the United States, "the potential of school arithmetic to unify elementary mathematics is not sufficiently known," Ma argues. "This is a blind spot for current U.S. elementary mathematics." Too often school arithmetic is equated with basic computational skills that require only inferior cognitive activity such as rote learning. Although many people in mathematics education view arithmetic as an ugly duckling -- that is, a collection of algorithms to be learned by rote -- Ma notes that "in the eyes of mathematicians it is often a swan" because of the mathematical structure mathematicians see in arithmetic. 
The New Math and the National Security State 
Ma: "How was the arena of arithmetic, as taught in other countries, abandoned in the U.S., and why? I believe that there must be some positive reasons that made it happen. However, serious reflections on this issue need to be conducted."
Dr. Ma mentions Sputnik as an event that triggered what would become the new math, an approach discussed in this book, A History of the New Mathematics Movement.  It was at the height of the Cold War that the many new employees within the burgeoning military-industrial complex first began to warn Americans of the need to compete with the Russians using other people's children. This new administrative superstructure flexed its powerful muscles and created many initiatives and programs. "The National Science Foundation had been established in 1950 with 15,000,000; however, after Sputnik, their annual fund was raised to 140,000,000." 
National Defense Education Act - Wikipedia, the free encyclopedia: "The year 1957 also coincided with an acute shortage of mathematicians in the US. The electronic computer created a demand for mathematicians as programmers and it also shortened the lead time between the development of a new mathematical theory and its practical application, thereby making their work more valuable. The United States could no longer rely on refugees from Europe to supply all of its needs (although this remained an important source), so it had to drastically increase the domestic supply. At the time, "mathematics" was interpreted as pure mathematics rather than applied mathematics. The problem in the 1950s and 1960s was that industry, including defense, was absorbing the mathematicians who should have been at high schools and universities training the next generation. At the university level, even more recently, there have been years when it was difficult to hire applied mathematicians and computer scientists because of the rate that industry was absorbing them.
Congress funded the National Defense Education Act (NDEA) (pg 13 at the link) in 1958 to the tune of one billion dollars (1958 dollars) in 1958:
National Defense Education Act of 1958 | Federal Education Policy History: "The National Defence Education Act (NDEA) also provided funds to state educational agencies for the purposes of improving the teaching of science, mathematics, and “modern foreign languages” (e.g., Russian, not Latin). Thus, the NDEA was the first major federal foray into K-12 curricula since the Smith-Hughes Vocational Education Act of 1917.
The massive amount of money that promoted STEM, as well as the strings on that money, changed the nature and role of public universities as well as the  lives of professors, emphasizing research over teaching.  Suburbs were expanding rapidly (leaving behind segregated poverty in the cities) and the deep centralization and expansion of school administration was underway allowing the changes to be easily implemented. The new math was a side effect of the growth of the National Security state or what Eisenhower famously called the military-industrial complex.

The massive effort did some amazing things: a man walked on the moon and I am typing this on a blog and posting it on the web. But the national security state and the corporation would consolidate their power in the neoliberal era. Neoliberalism would see US teachers reduced to fully-scripted lessons while police became constant in schools and the school-to-prison pipeline was fully formed. Even today, with unemployment, the push for STEM is unabated. Like vegetables force-fed fertilizer to support growth in depleted soils, the new corporate math texts are swollen and colorful but completely lacking nutrition, or coherent core subject content.

The administrative superstructure over teachers and schools now restricts teachers from learning and improving themselves (even if that is done by not providing support). That this administrative superstructure, like healthcare and finance, is a problem is clear. It creates the wrong incentives, removes the users from any significant feedback, and works against true education and knowledge by creating a credential industry that sells jobs.

A New Model of Elementary Mathematics: Numeracy Expands [1]
[1] Perhaps one long-term contribution of the more recent introduction of mass compulsory schooling in Asian countries is the expanded development of numeracy as part of alphanumeric literacy. Numeracy and the application of mathematical innovations grew steadily in the 20th century. How does this comparatively recent growth in numeracy show itself within the field of economics where a quasi-mathematical system subsumed the core subject structure of the field? How does mass schooling affect our consciousness: can we really skill-train endlessly with no side effects? 
It is also worth thinking about the trajectories of the development of alphanumeric literacy and phonetic writing in the East and West. The impact of alphanumeric literacy itself, much less mass education for extended periods of time, on human beings is still not understood nor is it easy to study.  For one, you need to have an agreed-on model of psychology to meaningfully discuss learning, and there is no agreement on a model at all nor will that happen soon. If economics is still trying to figure out if they are a science, psychology is even newer and few agree on the fundamental core subject structure.  In spite of that fact, the credential industry has given degrees in psychology for decades. (Full disclosure: I view analytic psychology as providing the best model available for work in the new discipline of psychology but I am in a small minority with that view. Jung wrote a good deal about this.)
In developing her critique, Dr. Ma presents a model of elementary arithmetic that is new. Unlike the new math, this model was developed by teachers. Dr. Ma describes the three further evolutions of elementary arithmetic seen in China: 1) the introduction of the four basic laws, 2) specific types of thought experiments, or word problems, as well as 3) pedagogical innovation (not only from China) that provides support for children to develop abstract thinking ability. The unique type of word problem used has a distinguished history that Dr. Ma describes:
These problem types came from ancient civilizations, such as Rome and China, and reflected an approach to mathematics different from that of Euclid. The theory of school arithmetic established by following the approach of the Elements emphasizes rigorous reasoning, but these word problems provide prototypical examples of quantitative relationships. Solving variants of these problems promotes flexible use of these relationships.  
This new model of teaching elementary mathematics, which allows children to do mathematics, instead of rote memorization, is an important contribution and it should be welcomed. Dr. Ma's work, and the teachers, students and schools she learned from and studied, have helped convey vital knowledge not only about math education in the US but also about numeracy itself. The beauty of elementary mathematics is not lesser than that of higher math though long years in hierarchical structures can bias many, I suspect.  Knowing the importance of arithmetic in developing numeracy can help in many ways other than this specific case. This doesn't mean that excessive drilling and competitive testing and ranking are good: the underlying assumptions about learners are deeply flawed. Those aspects of mass schooling are as pernicious in the East and as in the West.

This new model of teaching elementary mathematics is clearly far better than the dumbed down curricula that mass schooling produces, by its inherent structure. The fact that China, too, now has a strands approach shows that macro-level problems with the factory model persist and these problems are compounded in very large nation-states that face great inequalities, like the US and China. Finland may be able to reconfigure on a national scale but the US and China face far greater challenges and they must grasp a macro-level issue like the factory model to make effective change. 

Let's take a minute to think about this: Decades of over-processed curricula and top-down corporate control have completely lost the core subject structure of a vital subject.  This is an incredible indictment of our current system. The school system, as we know it, loses knowledge over time, in this instance, the loss was unnoticed for over 50 years. Are we sure other subjects actually contain intellectual substance or something worth learning? Can we all see that the emperor has no clothes?

As John Taylor Gatto warned, mass schooling truly dumbs us down. Another  core subject structure of mass schooling remains: confusion, class position, indifference, emotional dependency, intellectual dependency, provisional self-esteem, the necessity of surveillance.

The nation-state is a very new structure itself as is mass compulsory schooling in the factory model. What are the side effects of this mass increase of numeracy?  We can see the effects of mass regimentation using numerical ratings and ranking and its a growing problem.
Japan's Cutthroat School System: A Cautionary Tale for the U.S. - Noah Berlatsky - The Atlantic: "Again, though, that link between competitive schooling and Japanese triumph has broken apart over the last decades. In light of that, and of our own protracted ongoing experience with economic precariousness, it might be worthwhile for the U.S. to reconsider our current focus on schools as engines of economic attainment, either individual or national. Do we want all our students constantly rushing in a race to the top, even if, life being what it is, that top is sometimes not a mountain but a cliff? Is education entirely about succeeding economically? Or might there be other, more important kinds of success, involving connection, community, and rootedness? Both Japan and the U.S. could stand to think about whether we want to concentrate on getting schools to produce good workers, or whether we would rather have them help to make good human beings."
The US and China are both very large nation-states that have instituted mass compulsory schooling and making these large systems sustainable, humane and productive is proving deeply challenging. Neither nation, nor any other, has ever had an institutional system that filtered all children into all jobs. Even China's famous examination system, the progenitor of standardized testing, never provided an all-encompassing system for staffing the whole of society, a problematic goal of mass schooling.

Valuing teachers and ensuring they have the time and space to collaborate and the tools to learn with, is a huge lesson to the US. If teachers cannot learn and understand what they are teaching, they can't teach it to others. Teachers must have the time and space to collaborate, to develop and practice, to delve deepky not into tools or methods but into the actual subject matter they are teaching. The teacher's role within mass schooling in the US has been reduced over time to more of a guide. The role of the teacher in the East may carry far more depth than in the West and it is teachers who have developed this new model of teaching numeracy.

Part Two:  coming soon

down for the page count
coercion is a core function of schools
voluntary attendance
mass schools and the truancy trap
the compulsory attendance mindset
real school reform (and a changing view of attendance)

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