Curricular Coherence in the Age of Open Educational Resources
By Matt Larson, NCTM PresidentAugust 22, 2016
From its very founding, NCTM has actively promoted the use of high-quality curricular materials to support effective mathematics teaching and student learning. A critical feature of high-quality curricular materials is that they are coherent. Coherence, with respect to mathematics curriculum, generally means that connections are clear and receive emphasis from one year to the next, from one concept to another, and from one representation to another. High-quality materials are coherent pedagogically, logically, and conceptually.More than 15 years ago, NCTM enunciated the Curriculum Principle in Principles and Standards for School Mathematics (NCTM, 2000): “A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.” Fourteen years later, NCTM reinforced the importance of curricular coherence in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014): “An excellent mathematics program includes a curriculum that develops important mathematics along coherent learning progressions and develops connections among areas of mathematical study and between mathematics and the real world.” NCTM is certainly not alone in advocating curricular coherence. The authors of the Common Core State Standards for Mathematics identified coherence as one of their guiding principles and organized the content standards into clusters and domains that weave content together from grade to grade or topic to topic to make conceptual connections and coherence more obvious to teachers and curriculum developers alike.The increasing availability of online instructional materials—some of which are of high quality and some of which are not, and many of which can be downloaded at no cost—has added a new dimension to the curricular landscape for mathematics teachers and school districts. Some of the most engaging conversations about mathematics teaching today are taking place within online communities where teachers share instructional resources and ideas that they have either created themselves or found on their own online. A recent survey by the RAND Corporation found that the vast majority of math teachers, at both the elementary and secondary levels, reported they used materials that they developed or selected themselves to implement the Common Core State Standards for mathematics. There is no question that this practice is widespread.The dilemma is that while districts, schools, and teachers have greater access than ever to tools and resources for selecting and developing instructional materials, the skill required to develop a high-quality curriculum is both complex and often underappreciated. The widespread availability of online tasks therefore makes having and working with a coherent curriculum at the school and district level even more important because it is the curriculum that establishes the learning goals in a coherent progression and helps teachers see and understand the multiple pathways that students might take through the progression.NCTM itself has published online materials that provide examples of curricular resources that encourage teachers to integrate high-quality mathematical tasks and problems into their mathematics instruction. These materials stand as examples for teachers and schools in cases where the core materials may lack highly engaging, high-cognitive demand tasks or lessons. NCTM’s recent publication of exemplar Activities with Rigor and Coherence (ARCs) is an example of one such online resource. Each ARC is a series of lessons that addresses a mathematical topic and demonstrates the vision of instruction that Principles to Actions describes in detail. ARCs integrate a wide array of NCTM resources and include community features that offer opportunities for social interaction, feedback, and ratings.Ideally, teachers who select online instructional resources and engage in online community discussions would not be working in isolation but within well-developed professional learning communities in their schools. This sustained colleague-to-colleague communication would increase the likelihood of the selection of high-quality tasks that fit within mathematical learning trajectories and support the school and district’s curricular goals for students. Whether such collaborative task selection is feasible or not, the selection of online materials should be done in such a way that the instructional materials used in classrooms are situated within an overall coherent curriculum. That lessens the chance that students’ learning experiences devolve into a mere “collection of activities” rather than a coherent, well-designed curriculum.Stated very simply, the danger in online curricular selection is the undercutting of curricular coherence by the introduction of disjointed tasks that are of questionable quality, do not fit within the mathematical learning progression, and are incoherent. Perhaps the greatest danger is the potential for vast inconsistencies in instruction and highly variable learning experiences for students that in turn can lead to differences in student learning outcomes.Without question, curricular coherence is highlighted and enhanced when teachers work collaboratively and regularly with colleagues at the school level to plan instruction, implement the task, anticipate student work, respond to student learning needs, and provide consistency in curricular aims and instruction for students—no matter what teacher students might be assigned. Easy access to online tasks and communities makes the need to work collaboratively with colleagues in local professional learning communities more critical than ever before in the interest of safeguarding consistency in student learning experiences and outcomes.
AcknowledgementI would like to thank NCTM’s Emerging Issues Committee for its thoughtful work on a framing paper that was the basis for this President’s Message.
ReferencesNational Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: Author.
Coherence of the mathematics in the recourse of the calculator that in the excel sheet with formula is helper of the education. Subtraction in the adding of the multiplication is on rush-my-essay formula the main equal the bass of the curricular extra learning.
This is great news. I will write my thesis at https://paperell.com/write-my-thesis about it. I hope that it will help a lot of people with understanding the basis of education and why it's so important.
Better to have tried (something off the beaten path) and "failed" (the lesson-as-taught doesn't grab the brass ring of high levels of student engagement, participation, and insight/learning/growth) than to never have tried at all (thereby settling for the usual, generally dull and highly-directive activities of most commercially produced textbooks. Now, having probably mixed too many metaphors in an attempt to get the cliche to fit, let me suggest that the fundamental issue is never the materials but always the teacher. A teacher who downloads, finds in another book than the official text, pulls from the latest issue of one of NCTM's teacher journals, or takes from the colleague across the hall a lesson-as-written and mindlessly applies it to his/her own classrooms will generally do no better (and generally worse) than the original implementation/design of said resource. Garbage in, garbage out? Of course, the given lesson may be a good deal above the level of garbage, but it can't be a magical elixir, either, that miraculously works with every student in every class as taught by any teacher whatsoever. No such animal exists.
What's needed is both teachers who have been taught how to interrogate lessons in the light of their particular students, their own strengths and weakness as practitioners of the art and craft of mathematics teaching, and lessons that are offered up with reflection on the various "moves" within the lesson and the components thereof. That is, the person(s) posting these lesson frames need to draw back the curtain on how they came to craft them as they did and why they think doing things in a particular way makes sense, keeping in mind that it's unlikely that everyone, perhaps not even ANYONE, who is giving the proper reflection to such matters will agree 100%. And so there might well be reason to post "alternative hypotheses and conclusions" about how a lesson might be done, much like we expect from intellectually honest publishers of educational and other social science research. One size does NOT fit all, even if a limited bit of experimenting with one's own practice suggests that it does.
Excellent point. This is one of the reasons that I find NCTM's journals to be some of the best resources for planning: you can read why a lesson sequence or activity was designed in a certain way, and also how it worked with a particular teacher and group of students. Math teachers' blogs often give a similar context. In contrast, a typical textbook or other prepackaged curriculum has little or no reasoning or reflection included -- just a list of topics, procedures, and practice exercises.
That said, I appreciate the call for curricular coherence and wonder that no one has mentioned NCTM's Curriculum Focal Points document. Teachers at my school got together to study it and the NCTM standards over the summer. I found it very useful.
To some extent this is true: there is no "perfect" lesson, and it is possible to use even the best resources poorly. But this misses the crucial point that the converse is not the case: it is difficult for even a very good teacher to make a good, valuable lesson which challenges and develops students' thinking from a poorly designed resource/lesson without essentially rewriting it themselves. At least in the UK (where I live and work), most textbook exercises are very routine (procedural) and do not support the development of problem-solving or deep understanding (though this is beginning to change). I am currently working at Underground Mathematics, a UK Government funded project, where we are developing freely available teaching resources for 11th and 12th grade students; we aim to make them as robust as possible and provide guidance to teachers to help them get the most possible out of the resources. These resources are designed to provide students with the opportunity to develop their thinking and understanding. And of course we invite teachers to comment on how they have used them in the classroom so that other teachers can benefit from the "alternative hypotheses and conclusions" that you describe. It is possible for teachers to "break" the resources, but many of them are quite resilient to this. Take a look to see what I mean by this!
Thank you for starting this discussion. There are so many things to think about here.
In a perfect world, all teachers would have access to and use a well-developed and coherent curricula. Much of the movement to using online resources seems propelled by teachers' distaste for the curricula they have access to. But, I agree that an a la carte system of lesson selection creates real concerns about coherence.
You propose that collaboration between teachers can help to address the concerns about coherence. I agree, but I also think that there is another avenue that can be followed to improve coherence: make published curricula more appealing to teachers.
Many teachers seem to have a stigma against any traditional curriculum, but my experiences with some have been quite positive -- Connected Mathematics for example. However, I think there are many ways in which even the best designed curricula could be redesigned to be more appealing to teachers.
Digital - Many curricula offer digital products, but those products are often clunky and are not integrated into the actual curricula -- they are (or at least seem to be) produced afterwards by the publisher. Teachers desperately want modern digital tools built into their curriculum.
Better Hooks/Launches/Act 1s - Much of the appeal of online materials is that they have better lesson openers than traditional curricula. They have videos and setups that students (and teachers) find more engaging.
Open Problems - Teachers know how valuable these types of questions are, especially for engaging with the standards of mathematical practice. So, when they are not included in their curriculum, teachers have to work them in on their own.
Access - As the comments on this thread demonstrate, even when high-quality curricula exist, some teachers do not have access to them. The ideal curriculum would be free and downloadable for all.
Concise - Many curricula suffer from providing too many materials. They have lessons, tens of homework problems per lesson, projects, reflections, mid-chapter check-ins, pre-assessments, etc. Some of these materials are not high quality and the overwhelming amount of options makes it hard for teachers to know what is most important to use.
Simple and Clear Teacher Materials - Trying to go through the teacher resources for a curriculum can be a nightmare. There are often thousands of pages of materials split among multiple texts. A better curriculum would have much simpler lesson plans, chapter overviews, and explanations. That way, teachers could more quickly and easily see what the goal of each lesson is and why the lesson was constructed in the way it was.
A modern digital curriculum that teachers want to use is an alternative way to promote coherence. In addition to working collaboratively with teachers, I am hopeful that teachers will soon have access to a strong curriculum that is also more appealing.
I should very much like to connect with you. I've spen the last 3 years trying to develop the curriculum you describe. (Transparency: I work for Discovery Education; prior to that, I was at NCTM for 8 years running the Illuminations project.) I would love to share my experiences with you and hear more about your vision. I think that it very much aligns with mine. If you'd like to connect, I can be reached at firstname.lastname@example.org.
Very timely! And indeed timeless as your reference back to PSSM suggests. Honestly, I think this is a growing concern, given that:(a) Many existing textbooks really aren't that awesome (sorry to certain authors of those textbooks).(b) Teachers increasingly don't even have a textbook.
Having a community like #MTBoS that collectively vets tasks is definitely a great start... and having some way of curating those tasks would be awesome. But I still worry that this doesn't ensure coherence. Do the tasks have a common framing, using similar representations, notation, and so forth? So that they can hang together and build on each other?
So to pick on Prof. Lott's point, I wonder if NCTM could provide a backbone on which open source materials could be built. Serve as a convener of the process, perhaps with an editorial board to help ensure coherence. Of course, there would need to be some way of subsidizing this. Grant? Subscription fee? Major donation from a past NCTM president? Just brainstorming...
Coherence is a good thing, but it is not the only thing. Many teachers are stuck in an apparently coherent but pedagogically ineffective curriculum. Anxiety about coherence should not stop them from incorporating alternative activities or units into their courses, if that would result in deeper student understanding and/or improve students' attitudes and relationship to the discipline. Yes, online materials are uneven, but as far as I know, the most popular resources (nrich, Illustrative Math, Shell Centre, mtbos, Dan Meyer, NCTM, etc.) are reliable.
Teachers! If something in your textbook is not working, don't be afraid to try something else!
Thank you for your comments. I think we agree. As I pointed out, high quality on-line tasks do exist and you cited some of those. The issue isn't coherence versus quality. Clearly we need both. Neither a high quality task dropped into an incoherent curriculum, nor a low-quality task placed within a coherent curriculum will lead to deeper learning.
+1 Well put.
This piece is a great extension of Dan Meyers blog from a couple of months ago:
I agree that coherence is very important especially when considering Open Educational Resources. The #MTBOS has talked a lot about this and while work exists - https://emergentmath.com/my-problem-based-curriculum-maps/ this is of course not a complete set of curricular resources.
This quote struck me as interesting - "Ideally, teachers who select online instructional resources and engage in online community discussions would not be working in isolation but within well-developed professional learning communities in their schools." I feel like online collaboration tools such as Google docs would enable teachers to be able to expand the scope of expertise far beyond their specific school. Teachers working with ELL's can collaborate not just with their own staff but with others far beyond their geographical area. I do feel like a collaboration community beyond twitter could be facilitated by NCTM with the right tools. Not forums, something more social - heck, a nationwide Slack could be effective if grouped appropriately.
+1 nationwide slack channel
There are Math Educator/Education LinkedIn groups, Facebook groups, twitter groups, hashtags, edWeb forums, StackExchange, blog chats, subreddits... each their own echo chamber of sorts. The myriad non-profits, conferences, and math education researchers create other spheres of influence. Early career teachers are stuck asking, who can I trust? This kind of cross-talk is critical for our ongoing development as educators.
Thanks everyone for your work.
Thanks for your comments. I certainly agree that teachers can collaborate beyond their school and NCTM is in the process of creating a Curriculum Resources Collaboration Center as part of the NCTM website that will be facilitated by the Math Forum. I believe that in-school collaboration is critical to support equity. For example, if two teachers teach the same grade or course and each creates their own curriculum without collaborating, then one teacher might promote students' conceptual learning and problem solving, while the other might promote only procedural fluency. As a result, the learning opportunities for students in the same school vary widely.
I would love to work with a coherent curriculum, but haven't had that option since I started teaching in 2009. I'm sure many other teachers are in the same boat. I work in a district of about 50,000 students. The curricular materials we have pre-date Common Core by several years, so our district guidelines for 6th grade math have us doing the second halves of two formerly 6th grade books/units, followed by a unit from a now out-of-print source, followed by a 6th grade book/unit interrupted in the middle by a 5th grade supplement, followed by one that was designed for 7th grade... you get the picture. Obviously this is not the coherence the publisher was originally designing for.
Throwing out chunks of the curriculum in favor of Illustrative Mathematics and other tasks actually improves my course's coherence somewhat, but I really look forward to some day, probably years from now, when there will be funding for us to purchase an entire coherent, up-to-date curriculum or when a high-quality OER becomes available.
I do feel that Common Core is a huge improvement over our previous state standards, and I know NCTM has been in the lead of arguing for more up-to-date curricular materials and better implementation of the Common Core standards. You can't win 'em all. But without state and district support of Common Core, teachers have no reasonable choice but to roll their own & search the web. So I think providing ARCs, as well as guidelines and advice, is a smart and valuable move from NCTM.
Very nice piece. Now to get a coherent online curriculum, perhaps it is time that NCTM developed that curriculum completely. It is a gigantic costly task as discovered through the NSF materials. And it will have many critics but maybe the time has come.
I could not agree with you more. That said, I also feel like ANY curriculum will not solve issues related to pedagogy (Henri pointed this out below).
As a side note, the folks over at Illustrative Mathematics are currently writing a middle school (6-8) online (free) coherent curriculum that I believe will go public by the end of the school year.
Thanks Johnny. Your comment will certainly be something the Board will need to discuss. Matt.