What is at stake with Common Core mathematics and why is it so potentially harmful if it is just a minimum standard? In Part I of this series, I addressed commonly held beliefs concerning math education prior to the “reforms” of the early twentieth century that led to the math wars. In this post, I will address some of the motivations for reforms and offer some suggestions for parents and educators for math education. Even for those outside of the public school system, it is important to realize that there is so much money in “Common Core alignment” that the vast majority of mathematics publishing companies, including Singapore Math publishers, have begun to align their flagship curricula to this standard and ideology. The older curricula still exist, but publishers are not financially invested in them because the public school textbook market is the largest.
The roots of the Common Core movement began with flawed assumptions about human nature that can be seen in earlier mathematical reforms and have been compounded by the problem of the new minimum standards replacing normative expectations.
The first major “reform” in American mathematical education happened in the early twentieth century under the influence of progressives like William Heard Kilpatrick and John Dewey at the Columbia University Teachers College. Kilpatrick suggested in an address at the University of Florida, “We have in the past taught algebra and geometry to too many, not too few” (qtd. in Tenenbaum 105). His pragmatic reasoning was that learning the forms of thinking fostered by these disciplines not only wasted time and were not necessary for ordinary life, but could actually be harmful to those pursuing an ordinary life. He chaired the National Education Association’s committee on school reform where he argued that no mathematics should be taught that did not have clearly demonstrable “value.” In other words, traditional high school mathematics was to be reserved for a small, select group of privileged individuals. Before this reform, over half of high school students took algebra, but by 1955, this number had been reduced to a quarter of students.
Americans realized the limitations of this hyper-pragmatic approach when the Russians launched Sputnik in 1957. It became clear that Americans were losing the space race because the mathematical disciplines necessary for the sciences were not being taught. A coalition of math professors and teachers was brought together to create a “New Math” curriculum for American schools. As Morris Kline observed in Why Johnny Can’t Add, this New Math curriculum made some very necessary changes to the high school curriculum. The mathematical disciplines that are necessary and foundational for the study of the scientific disciplines again became standard curriculum: algebra, geometry, trigonometry, and in many cases, calculus.
Where this “New Math” went astray was in the elementary school curriculum. There were many factors involved. First, at that time, most elementary school teachers did not enter their field because of a love of math. They became elementary school teachers because of their love of children. Second, many of the college mathematics professors who helped in the curriculum revisions had a love of math but not much experience with how children grow or develop. They had difficulty distinguishing between the logical order of the mathematical subjects and the order that these mathematical skills and concepts ought to be taught. This situation was in large part due to a major change in the discipline of mathematics, which began to separate applied mathematics from pure mathematics and to the rise of hyper-specialization in the field of mathematics. Under New Math, for the first time in human history, pure mathematics became the goal of elementary education.
By the time Morris Kline wrote Why Johnny Can’t Add in the 1970s, it was clear that many American students were woefully underprepared for higher-level mathematics. Kline recommended that elementary mathematics should return to being aimed at the needs of all students, not just those who could become mathematicians. Furthermore, he pointed out that “knowing is doing” because multiple concrete experiences give students the foundations to unify these experiences into abstract understanding as they mature. In fact, as he points out, “premature abstraction may lead to sterility” because students do not have grounded reasons for understanding abstractions. He argued that the “informal thinking,” often disparaged as not rigorous by New Math, prepares students for more formal thought.
Furthermore, even in high school mathematics, the curriculum should emphasize both the mathematics used in the sciences (i.e., applied mathematics) and the historical development of mathematics rather than pure mathematics. In order to train the next generation of teachers, he recommended the creation of mathematics professorships aimed at the teaching of mathematics rather than the pursuit of mathematical research. While pure mathematics may be logically foundational to all mathematics, the human mind develops mathematically from the concrete to the abstract.
Many of the hallmarks of “New Math” eventually went out of favor, especially in elementary schools, but a progressive report made in 1980 by the National Council of Teachers of Mathematics (NCTM) started to emphasize “problem solving” rather than mental arithmetical calculation because new technologies like the calculator allegedly made mental mathematics obsolete. In 1989, NCTM Standards encouraged more emphasis upon conceptual understanding and problem solving in early grades rather than traditional rote learning.
While these standards were slightly revised in 2000, they did not differ in spirit from the 1989 Standards and became the philosophical principles underlying the 2006 Curriculum Focal Points and the 2010 Common Core Standards. One stated motivation for these standards was the desire to promote equity for all students. The goal was to create standards attainable in educational environments where teachers could not assume economic security, widespread literacy, and/or fluency in English.
Considered in light of the history of mathematical instruction, Common Core Mathematics (CCM) continues the New Math assumption that conceptual teaching of mathematics should be taught in elementary school while also implementing grade-level specific standards that serve as the goal of particular grades rather than the minimum benchmark that all students should reach. The issue is a failure to heed the words of Ralph Waldo Emerson, “We aim above the mark to hit the mark.” The new grade-specific standards are good for failing schools, but poor for schools that were previously doing well because there is a difference between the minimum one should expect of all students and what should be expected normally by most students under good conditions. When these two expectations are confused, standards fall for all but the failing students. For example, there is no CCM standard for working with money for kindergarten or first grade because money is now taught in second grade (but not reinforced in third grade). If these standards were seen as an end-goal, that would not be a problem, but they affect how much emphasis is placed upon a given skill at each level in new curricula, effectively “dumbing down” curricula that used to introduce money counting as early as kindergarten and first grade.
The so-called conceptual teaching of mathematics can be seen in the way the CCM standards mandate the teaching and practice of multiple strategies to solve simple problems. The first grade standard, for example, mandates “add and subtract within 20” while also insisting that this addition and subtraction be taught using multiple strategies:
First, it must be recognized that none of the strategies mentioned here are bad strategies, especially as students are learning their math facts. What is at issue is that Common Core curricula often insist that students become fluent in all of the ways a problem can be done all at one time, not allowing students to build upon previous knowledge. For example, even if a student knows how to skip count by 4, that student must also practice the two-step process of recognizing that 4 is twice 2, resulting in a problem that makes a student multiply by 2 and then add a double.
Educators widely recognize that students who must use and practice multiple strategies take up to three times longer to solve problems and that parents are not comfortable teaching strategies beyond the “standard algorithm” at home. They recognize the rise of math anxiety among students and frustration among parents. These difficulties, however, are seen as passing transitional problems that will eventually result in students who think mathematically rather than mechanically mimicking a set of steps to solve a basic equation. Because the goal is to foster mathematical thinking, it is not enough to help students by giving them a few strategies that work for them. Because math “fluency” is still a goal, educators assume that it will come, just later than previously. Unfortunately, the goal of fluency may be fundamentally at odds with the goal of teaching multiple strategies in early childhood.
One particularly strange aspect of Common Core reforms has been the absence of statistically significant studies about the relative effectiveness of mathematical curriculums that attempt to implement these reforms. In fact, very few major studies have been done on the relative effectiveness of top mathematics curriculums designed specifically to meet Common Core standards and criteria for assessment. The Federal IES Clearinghouse admits there are no studies that met evidence standards exist for Bridges in Mathematics, and Eureka and Illustrative Mathematics can’t be found in the database at all. Eureka has some small-scale studies that have mixed result (possibly due to COVID) that show potential effectiveness as does Illustrative Mathematics. Further studies also need to be done on pre-existing school curriculums that those who were aligned to the new standards (like Singapore Mathematics, Saxon Mathematics, and Math Mammoth). Even before the new alignment, the evidence for the effectiveness of these three curriculums is uneven: while Saxon has had studies that meet IES standards showing at least some effectiveness, Singapore Mathematics, despite its international acclaim, has not had significant U.S. studies. (Math Mammoth, popular among homeschoolers, would be much harder to assess for effectiveness.)
Given the widescale changes made by Common Core reforms, statistically-significant effectiveness studies need to be implemented, especially given the way limited studies such as those preliminary studies on literacy done by Dr. Arthur Gates at the Columbia University Teachers College have adversely affected American education when making widescale “reforms” in the past. While studies in individual curriculum need to be done, the aggregate of test results since the adoption of Common Core Math (and English) standards has been startlingly bad.
What these educators do not see is that those critical of the Common Core approach to math are not merely annoyed with having to change their ways, they fundamentally disagree with the developers about the nature of childhood development. There is no doubt that mathematics is, in fact, a language and that an understanding of that language, especially as it is necessary for applied mathematics, is the key to scientific advancement. The question is whether mathematics as a language ought to be taught in early elementary grades. Anyone with at least some experience with children, even the brightest of children, should recognize that they think in very concrete terms and that early introduction of concepts needs to be simplified and consistent with lots of repetition. While they can easily learn techniques for handling concrete problems, most young children do not learn and apply theories to concrete problems until they are much older. We see this even in the early introduction of language to children: The alphabet is taught with one song. The most common sounds of the letters are taught first followed by progressively more complex letter combinations.
Adults today worry that encouraging children in rote mathematics and concrete thinking will mean that they are unable to think abstractly as adults. Historically, however, the reverse has been true. Fluency with addition, subtraction, multiplication, and division facts along with concrete application in practical word problems gives children the foundation and confidence to look at ever more complex and abstract ideas. Before students can understand and fully articulate the why behind mathematics, they need the frequent experience of solving many practical problems, like handling money, cooking with fractions, measuring while building, playing board games, etc. It is one thing to give a child a few strategies, like counting-on or rounding, if that child is struggling; it is another thing to insist that all students practice all strategies rather than gaining mastery and confidence in standard algorithmic practice that gives consistent results every time.
Furthermore, it is not at all clear that mental and practical mathematics are no longer necessary in the age of the calculator, computer, and AI. Do shoppers get out a calculator when putting items into their grocery cart? How many car salespeople take advantage of customers who cannot do mental mathematics? How many people today regret not having learned about practical financial mathematics and statistics in school? How many physics students might benefit from being able to know the approximate answer to a problem before they calculate the precise answer? How many humanities students would be better at evaluating arguments if they understood some statistics? While AI may be able to automate many complex mathematical activities, is it wise for humans to lose this knowledge by not teaching mathematical skills to the next generation?
Historically, the best mathematicians and scientists for several millennia began with mathematical calculation before proceeding to the mathematical arts. Teaching calculation first will not hinder future mathematicians and scientists, but omitting mathematical calculation will absolutely harm students who go on to other careers. All students, even future mathematicians and scientists, need to know how to use mathematics in their everyday lives. While technology may change over time, the mental progression of the human mind from the concrete to the abstract remains the same. Elementary mathematics should not have “conceptual understanding” as its primary goal for early grades. Students need to begin by solving real world problems correctly and consistently. The more problems they solve (not the more ways they solve the same problem!), the more their conceptual understanding will grow.
