# What Does It Mean for a Curriculum to Be Coherent?

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.

# Quantity progression

The last remaining progression, the quantity progression, is here. Comments in the forums welcome!

# Catching up

The site was down for a few hours today because of a malware attack, but I think we have it fixed now.

I took the opportunity to catch up on comments in the forums; I was way behind! Thanks to all those who responded to readers’ questions. I will try to stay more on top of it. One of the things that has been keeping me busy is our work on grades 6–8 curriculum for Open Up Resources. It is being piloted this year, so that link is still password protected, but stay tuned!

Also, I am close to finishing up the Quantity Progression, the last one not yet done.

# Geometry Progression Grades 7 to High School

It’s been a while coming, but here is the draft Geometry Progression for Grades 7 to High School. As usual, please make comments and corrections in the relevant forum.

# Illustrative Mathematics Session at the Joint Mathematics Meeting

Illustrative Mathematics organized a special session at the Joint Mathematics Meeting on January 7, 2016 in Seattle, WA called Essential Mathematical Structures and Practices in K-12 Mathematics. Here is a description of the session:

The mathematics curriculum in the US has been shaped by myriad forces over the years, including the competition for market share among publishing companies, economic realities of school districts’ purchasing power, the ease with which teachers can deliver the material, traditional expectations of what mathematics classroom work should look like, and so on. Surprisingly absent from these forces is the nature of the discipline of mathematics itself. The focus of this special session was on identifying and describing the essential mathematical structures of the K-12 curriculum, as well as the key mathematical practices in the work of mathematicians that should be mirrored in the work of students in K-12 classrooms.

# Transformations and triangle congruence and similarity criteria

While we are all waiting eagerly for the geometry progression I thought people might be interested in this article by Henri Picciotto and Lew Douglas on a transformational approach to the criteria for triangle congruence and similarity. There is also lots of other good stuff on Henri’s transformational geometry page.

# Essays from the Noyce-Dana project: clarifying the mathematical underpinnings of secondary school

In 2008–2009 Dick Stanley and Phil Daro, with the help of Vinci Daro and Carmen Petrick, convened a group of mathematicians and educators to write essays clarifying the mathematical underpinnings of secondary school mathematics in the United States. At the urging of Dick Stanley I am publishing these essays here.

Some people are getting a message that comments have been closed if they try to post a comment on a post. I don’t generally close comments, so this is an error. I haven’t figured out how to fix it, but if this happens to you please send me an email.

# The confusion over Appendix A

A number of people have gotten in touch with me recently about Appendix A, so I wanted to clarify something about its role. States who adopted the standards did not thereby adopt Appendix A. The high school standards were intentionally not arranged into courses in order to allow flexibility in designing high school courses, and many states and curriculum writers have taken advantage of that flexibility. There was a thread about this on my blog 3 years ago, and there is a forum on the topic here.

Appendix A was provided as a proof of concept, showing one possible way of arranging the high school standards into courses. Indeed, on page 2 of the appendix it says:

The pathways and courses are models, not mandates. They illustrate possible approaches to organizing the content of the CCSS into coherent and rigorous courses that lead to college and career readiness. States and districts are not expected to adopt these courses as is; rather, they are encouraged to use these pathways and courses as a starting point for developing their own.

States will of course be constrained by their assessments. But Smarter Balanced consortium does not have end of course assessments in high school, leaving states and districts free to arrange high school as they choose. And although PARCC does have end of course assessments, they do not follow Appendix A exactly. See the footnote on page 39 of the PARCC Model Content Framework , which says

Note that the courses outlined in the Model Content Frameworks were informed by, but are not identical to, previous drafts of this document and Appendix A of the Common Core State Standards.

Furthermore, there are plenty of states not using either the PARCC of SMARTER Balanced assessments.

I hope this helps clear things up.