### Dodgy beginnings

In legal argument, every assertion cites authority: when lawyers know they are losing, they attempt to cloak weak arguments in language such as “it is clear that’’, glossing over the insufficient basis for why; strong assertions cite controlling authority, such as a prior ruling of the U.S. Supreme Court. The same citation requirements hold true for judicial opinions. The American common law system is grounded in its constitutions and legislation, but also on the principle of stare decisis, which means a strong legal opinion will cite another, preferably higher, controlling authority for coming down on one side or another. In the absence of binding authority, non-binding or persuasive authority is relied on: someone made an argument that won a case in another jurisdiction, the judge cites that decision and the law expands to a new jurisdiction.

Opponents of such decisions with weak legal precedent may deride them as “judicial activism’’, but judge-made law is a fundamental component of how our system works, and indeed, how the legal system has managed to survive. Of course, a judge may instead reject another non-controlling decision and cite an alternative argument for ruling differently. Thus, competing legal doctrines scatter like leaves in the wind until a higher court decides to consolidate and resolve contradictory rulings. It is often possible (and enlightening) to trace a winning argument in a high court ruling down through various lower court decisions and ultimately arrive at the original language source, which can be the unprecedented argument of a jurist publishing research (and personal opinions) in some obscure law journal.

Thus judge-made law, sometimes with questionable origins, becomes the law of the land and not always for the better. Toward the other end of the infallibility spectrum lies the scientific method, where studies confirm or refute hypotheses, and objectivity, transparency and replicability are the hallmarks of reliability. CCSSI boasts of its firm foundations: “the development of these Standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.’’ (p.4) When we first started this blog, we naïvely thought CCSSI’s language original; now we are discovering, in fact, that almost none of it is. As we analyze each of Common Core’s standards, we repeatedly ask ourselves: what is the underlying basis for the choices that have been made and where does the language come from? We’re certainly not the first to raise these questions.

Stanford University Professor R. James Milgram, who sat on the Validation Committee, expressed concern with a long list of CCSSI’s standards, writing that “[t]here are a number of standards…that are completely unique to this document’’ and “there is no research base for including any of these standards’’. Ideally, we would know from where and based on which research on its efficacy, each phrase, each standard arises, so that we could corroborate or attack the source. We are bracing for the worst: what if, in fact, the education pundits have issued mandates for math pedagogy based on dodgy research? We already suspect what we will ultimately find: the “studies’’ are actually individuals’ Ed.D. theses based on broad cognition hypotheses and corresponding latitudinal studies of limited numbers of children.

A central difficulty in our investigation is that, unlike in jurisprudence, original sources are not cited individually for each standard and prove difficult to trace, and it is becoming apparent that pieces from widely disparate sources were lumped together to form what is now called Common Core. This is the snarl we at ccssimath.blogspot.com are trying to untangle. CCSSI instead lists a “Sample of Works Consulted’’. When we started reading the end-referenced journal articles and other research, we were able to find some of the language and sample problems that provided the source material for CCSSI, but those too lacked specific footnoting, and also listed references at the end, apparently the accepted technique in such publications. Frankly, we are appalled with such weak referencing. Reading end-references sometimes led us to earlier iterations of the same language, which led to more references, ad infinitum. What we found in common, though, in every reference, was a plethora of vague, unsubstantiated language, mostly based on vague, unpublished educational research.

For once, we’d like to see the raw data of the actual research. One standard we have previously singled out for criticism is K.OA.3: “Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).’’ and its corresponding example in Table 1: “Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?’’ On our blog, we have rudimentary tools to analyze the searches that bring traffic to the site. Subsequent to the publication of that blog post, far and above the most common search sending us traffic is this standard, which we interpret to mean that kindergarten teachers are both trying to make sense of it and wondering how to implement it.

Readers of our blog know we don’t advocate posing a problem just because you can. Educators smugly confound students with some challenge and find self-satisfaction that at the end of the day, students can now solve it, but to what end? Perhaps in the linear progression underlying Piagetian cognitive development, any problem will suffice because you can see where you start and where you need to go, and you can easily ascertain (through the ubiquitous test, say) which students have crossed the threshold of competence, but true mathematics learning is not linear. How do we know that linear thinking pervades current notions of mathematics learning progressions?

Because educational circles give plenty of recognition to those authors. An influential pair of reports from the National Academy of Sciences, the 2000 “

How People Learn: Brain, Mind, Experience, and School: Expanded Edition’’ followed by the 2005 “

How Students Learn: History, Mathematics, and Science in the Classroom’’, claim to know “how the principles and findings on learning can be used to guide the teaching of a set of topics that commonly appear in the K-12 curriculum’’, specifically in our case, mathematics. One section of How Students Learn, written by Sharon Griffin, an associate professor of education and an adjunct associate professor of psychology at Clark University, begins:

After 15 years of inquiry into children’s understanding and learning of whole numbers, I can sum up what I have learned very simply. To teach math, you need to know three things. You need to know where you are now… You need to know where you want to go (in terms of the knowledge you want all children in your classroom to acquire during the school year). Finally, you need to know what is the best way to get there…

Were it so simple.

It is the pervasiveness of one-dimensional thinking of this sort that holds important “developmental milestones’’ that impedes effective mathematics curriculum reform. Now, this language may seem to mirror what we have been stating in this blog (see our blog post Concept of Area, Part 3, where we advocate “a well thought-out sequence that understands where things belong, understands where you are coming from and where you are going, and poses the right problems to foster the real thinking processes that we so strongly believe are the hallmarks of an effective education’’), but for several important differences. One, we are looking at math education from a 12+ year cycle, not one year. We want to instill not-easily-compartmentalized skills at an early age that will already be familiar, if not firmly established, and retrievable when the math becomes truly difficult. Griffin highlights a common fallacy of American math education, that a teacher only needs to know what is going on in the classroom that year.

How is the elementary classroom teacher with minimal mathematical skills going to handle the student that gains an insight that is years ahead of the rest of the classroom? Second, mathematics is not just about “acquiring knowledge’’; math at many levels is not necessarily as clean as one right answer, and those tensions can and should be introduced at a very early age. Everyone can be trained to go from point A to point B and a test can quickly check that, but the deeper understanding that comes with facing a dilemma cannot necessarily be measured. Seeing that math is not always black and white is an ability that education pundits themselves frequently lack; they don’t really understand the deeper mathematical connections and have no long term vision of an effective mathematics education. Returning to K.OA.3, a trainable, but rote task of questionable learning value, CCSSI actually points us to its origins, another NAS report, “

Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity’’ (National Research Council, 2009). Here is the source language, as it appears in the Mathematics Learning report:

In take apart situations, a total amount, C, is known and the problem is to find the ways to break the amount into two parts (which do not have to be equal). Take apart situations are most naturally formulated with an equation of the form C = A + B in which C is known and all the possible combinations of A and B that make the equation true are to be found. There are usually many different As and Bs that make the equation true.

And the grandma’s vase problem?

Put Together/Take Apart Situations

In these situations, the action is often conceptual instead of physical and may involve a collective term like “animal”: “Jimmy has one horse and two dogs. How many animals does he have?”

In put together situations, two quantities are put together to make a third quantity: “Two red apples and one green apple were on the table. How many apples are on the table?”

In take apart situations, a total quantity is taken apart to make two quantities: “Grandma has three flowers. How many can she put in her red vase and how many in her blue vase?”

These situations are decomposing/composing number situations in which children shift from thinking of the total to thinking of the addends. Working with different numbers helps them learn number triads related by this total-addend-addend relationship, which they can use when adding and subtracting. Eventually with much experience, children move to thinking of embedded number situations in which one considers the total and the two addends (partners) that are “hiding inside” the total simultaneously instead of needing to shift back and forth.

Equations with the total alone on the left describe take apart situations: 3 = 2 + 1. Such equations help children understand that the = sign does not always mean makes or results in but can also mean is the same number as. This helps with algebra later.

Even in these short excerpts from the report, several absurd generalizations pop out:

“…children move to thinking of embedded number situations in which one considers the total and the two addends (partners) that are “hiding inside” the total simultaneously instead of needing to shift back and forth.’’ They do? We certainly never thought about numbers this way. “This helps with algebra later.’’ It does? We’d like to see these hypotheses tested in a controlled longitudinal study. Although the report committee lists more than a dozen members, the lead authors were Doug Clements of SUNY Buffalo, Karen Fuson of Northwestern University and Sybilla Beckman of the University of Georgia.

These three authors also figure prominently in several of the other CCSSI source publications. Professor Clements’ educational background tops out with a Ph.D. in Elementary Education from SUNY Buffalo, Karen Fuson is professor emeritus of Northwestern’s School of Education and Social Policy, and while Sybilla Beckman of the University of Georgia is the only math Ph.D. of the lot, her research area stands out on UGA’s web site as “mathematics education’’, rather than a substantive area of theoretical or applied math. Individual emails to each of the three authors were unreturned. We don’t feel singled out for neglect, though.

Even Milgram “repeatedly asked for references justifying the insertions of these or similar standards…but references have not been provided.’’ This particular sections we cited, the entire report, and education reports in general illustrate a pervasive problem in education research: unfounded statements and the lack of scientific method. Such baseless statements appear all throughout these so-called education studies, then they are often taken for gospel because of the authors’ perceived expertise. Research methods that reach conclusions about what goes on in children’s minds based on observations of watching children at work would be laughed out of the scientific community; it’s inferences based on anecdotal evidence. Nonetheless, baseless conclusions form the justification for including “decomposition of numbers’’ in CCSSI’s kindergarten standards.

Not that none of Common Core’s references lack any substance. “

Informing Grades 1–6 Mathematics Standards Development: What Can Be Learned From High-Performing Hong Kong, Korea, and Singapore?’’, a study prepared by the American Institutes for Research, with the headlining author of Alan Ginsburg, long time and now retired Director of Policy and Program Studies for the U.S. Dept. of Education, the same Ginsburg referred to in CCSSI’s introduction, highlights “four key features’’ of the composite standards of “high-performing Asian countries’’. We refer the reader to the original text rather than try to summarize them here. We certainly agree with the sentiment against believing that “that merely replicating these composite standards is sufficient’’, but what we cannot find, though, is the adaptation to CCSSI’s goals of any of the composite features. Instead, we find the inclusion of standards with questionable beginnings. That puts CCSSI (and American mathematics education reform efforts) into the realm of wishful thinking, rather than basing itself on either hard data or emulating a proven success.

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Interesting!