One of the things I hope to research and better understand while I’m in graduate school is this: How do we create curricula that are more adaptable and flexible than the curricula represented in textbooks? I’m interested in the use of technology not only to transform the way that students explore mathematics, but also the way that teachers explore and plan from curriculum materials. The textbooks that I’ve worked on for a decade are wonderful resources — I do believe this — but they represent a static, one-size-fits-all approach that cannot adequately address the diversity present in classrooms and schools. I want to create versions of curriculum resources that help teachers make adaptations appropriate to their context.

I’ve been thinking a lot lately about how to begin exploring that idea. Most fruitful explorations, I find, begin with finding the right question to ask. In a recent conversation with one of my advisors^{1}, we agreed that an appropriate question to start this line of inquiry might be: In a mathematics curriculum, what are the things that a teacher might change without unintended mathematical consequences, and what are the things that must remain in place?

As I thought about that question, it seemed to me that it boiled down to simpler question: What are the essential features of a curriculum?

Or even more simply: What is a curriculum?

Simple as it may seem, I’ve been working through an answer to this question for years. Many times, I’ve done a thought experiment, trying to make a list of essential features of a curriculum, hoping to get a better handle on the term. I have consistently come back to the same four features.

First, there’s *completeness.* Several dictionary definitions of *curriculum,* including this one from the Cambridge English Dictionary, specify that a curriculum is *all* of something: all courses taught in a program, for example. Thus, a curriculum must be, by some definition, complete. This also implicitly justifies a related, second feature. Because a curriculum must be complete, there must be a way to define that completeness. This means curricula must be *in reference to something.* There are *subject-specific curricula* and *school curricula* and *second-grade curricula,* for example, but never just *curricula *with no referent.

A third feature comes to mind when looking more deeply at the mathematics curriculum literature. As a professor^{2} recently pointed out to me, mathematics education scholars have become increasingly more careful about adding meaningful adjectives before the word curriculum. Remillard (2005), for example, distinguishes between the *written curriculum,* which is generally represented in curriculum guides or textbooks, the *planned curriculum,* which is the plans teachers create from examining the intended curriculum, and the *enacted curriculum, *which is what actually happens in classrooms. The inclusion of *planned* and *enacted* curricula seems to me a pretty clear indication that a curriculum is more than a list of things students learn. A curriculum includes some attention to *what they do* in order to learn. For lack of a better term, I call this third feature *activity.*

To recap, so far, we have established that a curriculum is a *complete* representation of what students learn in *reference to something particular,* with attention to the *activities* they complete to achieve that learning.

These three features feel reasonable and useful, and I’ve been tempted at points to accept them as sufficient. But always I come back to one idea that bothers me: If those three features are sufficient to define a curriculum, then a curriculum is no more than a collection of activities. One creates a curriculum for a *particular thing* by gathering *activities,* continuing to add more until coverage is *complete.* These three features give no attention to the relationships among the activities, such as how they are ordered and why they are ordered that way.

There is a fourth feature, then, that captures the ways in which a curriculum is greater than the simple sum of its activities. A term often used in the literature to refer to this is *coherence* (e.g., Ruthven, 2016; Yerushalmy, 2016), though admittedly it is as ill-defined as curriculum in many cases.

So, by my definition, a curriculum is:

- A
*complete*representation of what students learn; - In
*reference to something particular;* - With attention to the
*activities*they complete to achieve that learning; - And the
*coherent*manner in which the activities are related.

These are the four features of curriculum to which I always return in this thought experiment. For comparison, consider the following quote shown on the website for the Center for the Study of Mathematics Curriculum (CSMC^{3}): “Mathematics curriculum: what is important, what is expected, how it is organized and sequenced, how it is taught, and what students learn is the core around which mathematics education revolves.” (CSMC, n.d.)

What do you think? Do the two capture the same ideas? I think so. Because CSMC is concerned in particular with mathematics, that captures the feature of curricula being in reference to something in particular. I see “what is important, what is expected” as similar to the notion of *completeness, *“how it is taught and what students learn” as similar to the attention to *activities,* and “how it is organized and sequenced” as a particular way of thinking about *coherence.*

Not a perfect mapping, certainly, but a reasonable one.

If these four core features of a curriculum hold up to further scrutiny, I think they could be a useful framework for thinking about the ways curriculum designers might make curricula adaptable without losing what makes them a curriculum, particularly if the curriculum is delivered digitally. A digital curriculum could allow for adjustments, and the core features could be used to put certain limits on those adjustments. *Completeness,* for example, could be maintained by defining some limits on what could be removed from a given curriculum. Likewise, the idea that curriculum must be in *reference* to something could be maintained by limiting what could be added. *Activities* could be maintained simply by making activities the manipulable unit — i.e. making activities the “things” that teachers can move around and change — and *coherence* could be maintained by defining relationships among the activities that limit the ways in which the activities can be manipulated (e.g., preventing one activity from being moved before another).^{4}

While I’m not a fan of framing adaptations in terms of the limitations put on them, I do so here for lack of better vocabulary for describing it, and keeping in mind that such a system will have the overall effect of removing many of the limitations of static print curricula.

What do you think? For the curriculum developers out there, what do you think are the essential features of a curriculum, and how could we embed tools in digital curricula that maintain them?

For the teachers out there, what kinds of adaptations do you make to your curriculum materials now, and how would you like curriculum developers to better support you in doing so?

**References**

Cambridge English Dictionary. (n.d.) *Definition of curriculum.* Retrieved on 08 September 2017 from http://dictionary.cambridge.org/us/dictionary/english/curriculum

CSMC. (n.d.) *Center for the Study of Mathematics Curriculum: An NSF Center for Teaching and Learning.* Retrieved on 08 September 2017 from http://www.mathcurriculumcenter.org/

Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. *Review of educational research*, *75*(2), 211-246.

Ruthven, K. (2016). The Re-Sourcing Movement in Mathematics Teaching: Some European Initiatives. In M. Bates & Z. Usiskin (Eds.). *Digital Curricula in School Mathematics* (pp. 75-86). Charlotte, NC: Information Age Publishing.

Yerushalmy, M. (2016). Inquiry Curriculum and E-Textbooks: Technological Changes that Challenge the Representation of Mathematics Pedagogy. In M. Bates & Z. Usiskin (Eds.). *Digital Curricula in School Mathematics* (pp. 87-106). Charlotte, NC: Information Age Publishing.

Notes:

- Credit to Dr. Jack Smith at Michigan State University for getting me started on this line of thinking.
- Credit goes again to Dr. Beth Herbel-Eisenmann at Michigan State University.
- While looking up the CSMC’s description of curriculum, I realized that the center was a collaboration between four institutions, three of which I have some affiliation with: Western Michigan University, where I earned my B.A. in mathematics, the University of Chicago, which was my professional home for ten years, and Michigan State University, where I’m currently pursuing my Ph.D. I’m newly grateful that my path has taken me to places where I have been able to learn from pioneers and leaders in my field.
- Many of the ideas in this paragraph stem from conversations I had with my colleague Jim Flanders during my time at the University of Chicago, while we collaborated on an effort to convert the
*Everyday Mathematics*curriculum to a database.

I like the ones you’ve identified. I think a modern definition would need to include some UDL concepts. It’s not a static – here’s what they should learn. Instead, it’s identifying the core ideas and ways to provide learning that matches the capabilities of the learner. I’m not sure what the right way is to phrase this, but it seems there should be some attention to tailoring to the learner.

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Yes, I agree there needs to be some attention to that. There’s a lot of unpacking to do particularly in the Activities and Coherence elements. I think your comment relates most strongly to the Activities element. It’s an issue of identifying what is core or unchangeable about the activities (commitment to a certain pedagogy, for example) and what is adaptable according to context — and according to whose point of view. Your comments focus more on the adaptability part — so important and so tricky.

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Hi Katie!

I’m enjoying your blog. Have you checked out Youcubed by Jo Boaler and team at Stanford? They are developing a curriculum, though one designed to support/supplement standards-based curricula, and I just read their short paper about their approach: http://www.youcubed.org/wp-content/uploads/2017/08/What-Is-Mathematical-Beauty-1.pdf

I’d like to order the 4th grade book to see more. Curious about your thoughts!

Amanda

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Hi Amanda! Thanks for reading!

I haven’t looked at this part of the YouCubed project before. I’ll check it out.

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