If you are a member of NCTM (or just on their email list), you probably received an email linking to Matt Larson's new blog post: "Curricular Coherence in the Age of Open Educational Resources." I found this article fascinating; it addressed so many of the things I have been wrestling with but also left me with some lingering thoughts. Let's run through some of the main points:

"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."

It is hard to argue with this opening statement. We all know that students benefit from consistency as well as well-planned and organized learning experiences. When each lesson builds on prior knowledge and experiences, students are more likely to learn and understand mathematics. Understanding is building networks of connections between concepts in the brain. Coherence definitely supports that.

"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."

"Online instructional materials" is such a general term with lots of different materials under its umbrella: teachers-pay-teachers, desmos, auto-generating worksheets, blog posts, Khan Academy, Illustrative Mathematics, Three-act-math, WODB, Estimation 180, etc. There is a huge variety of quality and style here. I worry that lumping them together is going to muddle the point for many teachers. If you use any of these resources and feel like you are being criticized, it is so easy to get defensive and shut out new ideas.

"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."

#MTBoS shout-out! Definitely agree with this statement that "engaging conversation" is happening online. However, #MTBoS benefits from being a self-selected community. The teachers that participate in it have opted-in. This greatly increases the likelihood that new members subscribe to beliefs that are similar to the overall community's. It's easy to have productive discussion when most participants are on the same page to begin with.

In contrast, a colleague that has been assigned to teach the same course will not necessarily hold the same beliefs as you. This is a great opportunity for growth but is probably less likely to produce productive discussion unless protocols are put in place like professional learning communities.

"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."

From my anecdotal experience, it seems as if a very small group of teachers "use" a curricula. By "use" I mean follow the day-to-day lessons and activities set forth by the curriculum (with few exceptions).^[3]To be clear, I am NOT saying that when you "use" a curriculum you never integrate additional lessons or activities. Just that it is NOT the norm and takes up a small percentage of your class time.

Some of this is caused by lack of access to good curricula, but much of it is not. I have encountered stigma against using curricula. It seems that there is a belief that only beginner or lazy teachers use curricula, and that the best teachers always create their own materials. It is almost a hipster-type mentality, once a curriculum is published and goes mainstream, it cannot be good. That seems insane to me. Every curriculum has its strengths and weaknesses, but some of them are REALLY good. It does not feel right to dismiss them all off-hand.

"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."

This is the most interesting paragraph in the whole letter. I completely agree with his statement that "the skill required to develop a high-quality curriculum is both complex and often underappreciated."

Think about how you spend your time as a teacher. Mine looks roughly like this:

• Actively teaching or working with kids: 50%
• Email, grading, copying: 30%
• Lesson design and preparation: 20%

Curriculum developers are spending 80% or more of their time developing lessons and have years to develop a curriculum.^[4]That 80% figure is a guess, but I feel confident that curriculum developers' percentage is significantly higher than the percentage of time I spend developing lessons. Also, do not forget that they can spend years developing a curriculum whereas a teacher has to be ready for day 1 of school. If only 20% of my work time is dedicated to developing lessons, how can I possibly produce work of comparable quality consistently. I can make a few great lessons here and there, but a day-to-day program is going to be near impossible.

The answer is that I can work collaboratively with peers (in my school or over #MTBoS). That's what's happened with things like Geoff's Curriculum Maps; but, even with how well those are thought out, coherence is still a real concern. There are so many gaps and inconsistencies in those maps. That's not a criticism of Geoff, it's just an inevitable result of compiling lessons from so many different sources that are not designed to build on one another.

But, I think Larson stops short of what he actually wants to say here. He loves the active discussion happening over #MTBoS (with good reason) but does not like the trend of going away from published curricula towards collections of lessons. Even when those lessons are individually strong. His thesis is summed up in the quoted paragraphs below: given that teachers are moving away from published curricula, it is crucial that they are extremely thoughtful in how they build their own curriculum.

Personally, I think this is impossible. We already struggle to find the 3+ million qualified teachers we need to teach our students. It is not possible to ensure that all of those teachers are trained well enough to individually (or in small groups) select tasks to build their own curricula.

"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."

"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."

This is the politest way of saying that when teachers use different self-generated and self-organized materials there is going to be significant differences in the quality of their teaching. With so many teachers developing their own curricula or combining materials to form something like a curricula, some students are going to get lower quality instruction.

"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."

Collaboration is great, but I worry that it is not enough to overcome the issues with self-curated curricula. Developing cohesive curricula takes too much time and care to be done while also teaching and grading and meeting with parents and supervising recess and doing all of the other millions of things that teachers need to do.

But, even if you agree with me, there is still the issue that many many teachers are building their own curricula. One reason that is happening (I think) is that teachers are not getting what they want from the curricula they have to work with at their school. So, how can curricula be more appealing to teachers so they get used more. I have some ideas:

• Digital - I have yet to see a curriculum that really embraces the digital world. Many now offer digital portals, practice problems, and pdf copies of the texts, but those resources are clearly built AFTER the curriculum is designed. They are not integrated into the actual designs. A modern curricula would be designed from the ground up to be a digital medium. Videos would be integrated into the lessons. Students would be using resources like Desmos, Google Docs, and GeoGebra as part of the actual lessons. Teachers are driven to use their own materials because they see the benefits of these digital tools and do not want them to be slapped on to the textbook lessons.
• Better Hooks/Launches/Act 1s - The hook is the question or situation that sets up a lesson. It creates the need to investigate and learn some mathematics. Much of the appeal of the teacher designed online resources is that they have great hooks. Most traditional curricula have unengaging set-ups. They say things like "Two students in Mr. Schwartz's class are arguing about whether..." or "Sometimes business owners need to calculate tax..." These are hardly inspiring.
• Include More Open Problems - Many curricula lack open questions. Every problem or prompt has an obvious answer and strategy for solving. They especially lack open-beginning and open-ended questions. Teachers know how valuable these types of questions are -- especially for engaging with the standards of mathematical practice -- so they have to work them in on their own.
• Access - In an ideal world, a great curriculum would free to download. Is there no model for which this can happen? Couldn't a group of designers self-publish? Could we give grants to pay for the development so that the end result could be free for every teacher in every school? I do not know enough about curriculum design to answer these questions. But I suspect it is possible and that the publishing companies would REALLY dislike it.
• Concise - Many curricula I have used suffer from being too full. They have writing prompts, side-projects, investigations, practice problems, more homework than you could ever assign, check-in quizzes, pre-assessments, grading rubrics, additional practice, lesson plans, quick lesson overviews, and on and on. It would take most classes 3+ school years to do everything provided with a curriculum. So teachers start skipping things and that undermines the coherence of the curriculum. I realize that a lot of those materials are included because they help to sell the curriculum, but I think many teachers would be happier with something that had the fat trimmed off of it.
• Simple and Clear Teacher Materials - Trying to go through the teacher resources for a curriculum is a nightmare. There are often thousands of pages of materials to read. And, while the details you want are often there, you do not have the time to find them. A better curriculum would have much simpler lesson plans, chapter overviews, and explanations. That way, teachers can more quickly and easily see what the goal of each lesson is and why the lesson was constructed in the way it was.

Math education needs to get better, but I do not think the answer is for millions of teachers to develop their own curriculum. There are obvious benefits to working with established curricula, but curricula also has to step into the modern age and better meet the needs of teachers and students. We can get there, but there is still a lot of work to do.