Al Cuoco and I have been thinking about this question and have developed some ideas. I want to write about the first and most obvious one today, the principle of logical sequencing. I’ll write about others in the weeks to come.
Remember the distinction between standards and curriculum. While standards might remain fixed—a mountain we aim to help our students climb—different curricula designed to achieve those standards might make different choices about how to get there. Whatever the choices, a coherent curriculum, focused on how to get students up the mountain, would make sense of the journey and single out key landmarks and stretches of trail—a long path through the woods, or a steep climb up a ridge.
By the same token, mathematics has its landscape. CCSSM pays attention to this landscape by laying out pathways, or progressions, that span across grade levels and between topics, so that a third grade teacher understands why she is teaching a particular topic, because it will help students with some other topic in the next grade and build on what they already know.
This leads us to the first property of a coherent curriculum: it makes clear a logical sequence of mathematical concepts.
Consider, for example, the concepts of similarity and congruence. It is quite common in school curricula for similarity to be introduced before congruence. This comes out of an informal notion of similarity as meaning “same shape” and congruence as meaning “same shape and same size.” However, the fact that the informal phrase for similarity is a part of the informal phrase for congruence is deceptive about the mathematical precedence of the concepts. For what does it mean for two shapes to be the same shape (that is, to be similar)? It means that you can scale one of them so that the resulting shape is both the same size and the same shape as the other (that is, congruent). Thus the concept of similarity depends on the concept of congruence, not the other way around. This suggests that the latter should be introduced first.
This is not to say you can never teach topics out of order; after all, it is a common narrative device to start a story at the end and then go back to the beginning, and it is reasonable to suppose that a corresponding pedagogical device might be useful in certain situations. But the curriculum should be designed so that the learner is made aware of the prolepsis. (Really, I just wrote this blog post so I could use that word.)
Although the progressions help identify the logical sequencing of topics, there is more work to do on that when you are writing curriculum. For example, the standards separate the domain of Number and Operations in Base Ten and the domain of Operations and Algebraic Thinking, in order to clearly identify these two important threads leading to algebra. But these two threads are logically interwoven, and it would not make sense to teach all the NBT standards in a grade level separately from all the OA standards.
In the next few blog posts, I will talk about three other aspects of coherent curriculum: the evolution from particulars to deeper structures, using deep structures to make connections between topics, and coherence of mathematical practice.
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With all due respect, Bill, I think you need to explicitly recognize just how complex a task building a curriculum is based on one set of standards that is inherently viewed as prescriptive and which is pre-loaded with dire consequences for not staying on THE straight and narrow pathway from point A to point Z. Everyone who is “ruled” by national (or state) standards is likely to see that whatever the sequence of topics is as presented in those standards is the One True Path that must be followed. To diverge is to court dangerous consequences from the powers that be.
That perception might not have been the intent of those drafting the standards, but it would be naive and/or irresponsible not to anticipate how things inevitably would play out in practice given the past track record of other major standards efforts in mathematics and literacy in this country. Of course, there is also inevitable skepticism, criticism, and resistance to standards, but for the majority of teachers, administrators, and parents, the perception is that the Common Core was written to be followed like many districts expect scope and sequence calendars to be followed (I have seen this in Detroit, to name one large district where high school teachers fear diverging from the district’s calendar lest they be “caught” and punished). This sort of thing kills creativity on the part of teachers and students, and it absolutely undermines the ability of instructors and students to take advantage of serendipitous opportunities to explore and play in the rich fields of mathematics.
No matter how coherent a curriculum might be based on standards that themselves are, ostensibly, coherent, they are only coherent from one point of view: that of their author(s). It is folly to believe there is some privileged objectively universal stance from which one can glean the perfect ordering of topics that will serve all learners well. All a good curriculum can hope to do is to provide some framework, preferably one with a variety of pathways and branchings, never with tight strictures against finding sound alternatives to the choices provided, and indeed offering encouragement and suggestions as to how and when to leave the “official” pathways.
That is not to suggest that all viewpoints are equally good, all choices of paths equally likely to be effective, all possible journeys equally worth pursuing without limit. We always have to make sensible evaluations as we begin to pursue alternative routes. But this is precisely where teachers’ pedagogical content knowledge comes into play. And I fear that the combination of federal strictures and monetary threats intended to ensure that educators stay within a very narrow distance from some intended pathway, coupled with publisher guidelines codified in official teacher manuals, and the tendency of administrators and many teachers themselves to be lazy, fearful, or poorly-prepared to make the necessary countless decisions about teacher moves that arise in the course of lessons means that in spite of the best intentions of everyone involved in mandating, crafting, disseminating, and implementing any set of standards or turning them into a “coherent curriculum,” I believe that in this country, at least, they will have a conservative or reactionary effect upon what mathematics is taught and how lessons are presented. And that is in spite of the one feature of the entire Common Core State Standards that I find admirable, the Standards for Mathematical Practice. Some lip-service is paid to these by publishers and administrators at various levels of the federal, state, and local educational institutions, but the focus from the beginning has been and inevitably will continue to be the content standards because of high-stakes testing and its consequences. And until the culture of testing and assessment changes dramatically, the coherence of standards and/or curriculum matter far less than we might believe or wish.
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