Prompted by NCTM's call for "grand challenges,"
in my last post I looked back at Hans Freudenthal's 1981 "Major Problems" paper. We've made progress in the past 30+ years, and we should recognize that. But that doesn't mean other challenges don't await us, and in this post I'll look at some suggestions made by some fellow bloggers. If this looks like "armchair challenging" it's probably because it is, rambling commentary and all.
Before I continue, it's worth noting that all four bloggers I found writing on this topic are white males. (And I am, too.) If this doesn't bring to mind a grand challenge for the future of math education, I don't know what should.
Robert's first suggestion is to develop an open curriculum for high school and early college. Sure, we've had many curriculum projects, but I can't say I've seen many that try to seamlessly span high school and college. It makes me realize that textbook companies typically package things in ways that align with the jurisdictions of district decision-makers, but there's really no reason it has to be that way.
We currently have some open curriculum projects that might give us a start on this challenge, such as the
Mathematics Vision Project out of Utah and the
EngageNY materials from New York. I say "give us a start" for two reasons: neither set of materials are very mature (and thus quality can be suspect) and such a project should plan for the evolution and improvement of the materials over time.
Side story: I was having dinner this summer with a retired mathematics education professor and she was telling me about her experiences volunteering to help tutor kids at a local high school. Our conversation went like this:
Her: "I didn't recognize the materials they were using, but they're a mess. It's something they found online and I don't know who put it together, but it looks like different people wrote adjacent lessons and never talked to each other, because there were big jumps from one topic to another with no explanation."
Me: "Let me guess. Are the materials from New York?"
Her: "No, Utah."
Me: "That was my second guess. And your guess about different people writing different lessons without much coordination is a very good guess of what probably happened."
Robert's second and third challenges involve the creation and use of concept inventories for mathematics, like the force concept inventory (FCI) for physics. I
hear this get discussed occasionally and I'm aware of some efforts for inventories in calculus and statistics, but they aren't nearly as well recognized or used as the FCI. What's the advantage of having these inventories? They tend to make for great pre-post tests for a course or to judge if a particular teaching approach is better for students' conceptual understanding. Last week I attended a talk by
Stephen Pollock who talked about his work in physics education research and the improved results we're getting in CU's physics program. The FCI played a key role in that progress, as it allowed professors to self-monitor their courses and compare their results to others who were attempting to improve their teaching. These kinds of standardized assessment tools could be equally useful and powerful in mathematics departments, especially when used in a self-monitoring sort of way instead of the all-too-common external-and-top-down-accountability-enforcing sort of way.
Robert's last recommendation is to have a preprint server for math education research. As he notes, this is a road we've tried to go down before and we didn't get very far. I don't think the problem has nearly as much to do with policy or categories of the
arXiv as it does with the lack of a "preprint culture" in mathematics education. What I learned in those previous preprint discussions, and in my observations as a developing scholar, is that math educators regularly and happily share work in progress —
with a select group of people. In math ed, there doesn't seem to be widespread faith in anything like
Linus' Law, the open source software dictum that says, "With enough eyeballs, all bugs are shallow." I think the math wars led to a lot of distrust, and some of it is very rational. It's safer to only share preliminary work with a few scholars who share similar methods and theoretical frameworks, and then refine the work after peer review before publication in a journal whose readership is likely to understand the work. Maybe it shouldn't be this way, but to move forward we're going to have to confront some of these beliefs.
Patrick described in some detail a single grand challenge: "Build and maintain a free, comprehensive, modular, and adaptable repository of learning materials for all secondary mathematics content." It's worth reading his post and the comments. This challenge hits close to home for me because it touches on my own research, including the difficulty of coordinating distributed curriculum development and the infrastructure needed to support the customization of curriculum.
I've always been intrigued by the concept of "modular and adaptable" curriculum materials. Personally, I thought I did my best work as a teacher when I
offloaded my curriclum to a high-quality textbook that I'd been trained to use. That's an anathema to many math teachers who take improvisation of curriculum to be a sign of quality teaching. (It's not, by the way. There can be good and bad improvisation, just as there can be good and bad offloading.) I tried writing my own curriculum for a while and found it exhausting and ineffective. In a couple hours per day, I just couldn't create from scratch anything that I thought was as good as the texts coming from university-based curriculum teams with decades of experience and millions of dollars of funding. Go figure. I got better results when I leveraged the rigor and coherence of a text that integrated topics, contexts, tools, and routines across its lessons and units.
With enough effort, however, Patrick's recommendation could lead to a set of materials that are both modular
and coherent. I've always seen these in opposition, a sort of "textbook paradox." I speculate that teachers who value being able to adapt and improvise with their curriculum will resist or find ineffective those textbooks built around coherence. It's relatively straightforward to replace a lesson in a very traditional textbook that relies on an isolated set of examples and practice problems. But for reform-based materials, such as
IMP,
CPM, and
Everyday Math, skipping around in the textbook can lead to trouble.
Saxon texts, for that matter, with their use of "incremental development," should make a teacher think twice before skipping or improvising a lesson. Thus, the paradox: teachers who want to improve the quality of their curriculum materials probably have an easier time adapting materials that are lower quality to begin with, but if they start with higher-quality materials, adaptation can sacrifice coherence and make adaptation more difficult.
Adaptation can still be done with any curriculum, but it takes skill. Currently, that skill must come almost entirely from the teacher, as the texts aren't smart enough to know what you've been skipping. Take Patrick's challenge far enough, however, and maybe we could have a curriculum that
is smart enough to know what you've used and not used. Imagine a statistics curriculum that automatically modifies tasks to use a preferred data set, or a system that reminds you that you should probably include a lesson and practice with mean absolute deviation prior to teaching standard deviation. Or, for algebra, imagine a system that let you decide whether to teach exponential functions before or after quadratics, with the curriculum being smart enough to recommend appropriate modeling tasks. When I helped a school pilot
Accelerated Math in 1999 and used the exprience as my student teaching action research project, I really thought we were on the cusp of a wave of "smart curriclum" that would help build coherence into teacher-adapted curriculum. We're not there yet, but a challenge like the one Patrick describes could get us much closer.
David's grand challenges focuses more on people than materials: "Develop a comprehensive, national professional development model that supports the high quality mathematics instruction they have been promoting for many years." ("They" refers to NCTM.) David breaks this challenge into bullet points around the development and scaling of "core practices."
I'm a firm believer in this idea. I get resistance from those who love the creative and spontaneous aspects of teaching, but I think that learning to teach should involve the learning and practicing of key teaching practices. Thankfully, there are some very good people working in this area. Until recently, their efforts were somewhat scattered and referred to with such names as "high-leverage practices" or "ambitious teaching." Thankfully,
at AERA this past spring, many of the heavy hitters doing this work came together to address the need for a common language around these practices and supporting their development and use. For a good idea of what a list of core practices might look like, check out the
Teaching Works project from the University of Michigan. I have a hard time finding anything on that list that doesn't seem essential to quality teaching, and it reminds me that the list is really the easy part. The real work comes in developing those practices in preservice and inservice teachers, and I'm glad that David had his mind on that development when he articulated his grand challenge.
Bryan Meyer:
Bryan's challenge isn't math-specific but it could help a lot of math teachers. Our expectations for teacher collaboration exceed our opportunities, and changing this involves a lot of people and resources. In some countries there are limits to how many student contact hours a teacher can have because they are expected to be collaborating with or observing other teachers for several hours each day. What if we did that in the United States? We'd have to seriously rethink our resources. Suppose you currently teach six periods a day with about 24 students in each class. What if you only taught four periods with 36 students in each class, and you had the extra two periods to work with other teachers to ensure your instruction in those four periods was better? (For those of you who already have 36 students in your classes and are working out even larger classes in your heads, I'm sorry.) Or, instead of changing class sizes, what if salaries were lowered to accommodate the hiring of extra teachers?
While these questions suggest difficult choices, they do seem like questions that could be answered with adequate research, and maybe there exists some research already that could help us answer them. Still, research in education isn't always very effective at changing school cultures or how resources are allocated. I don't want to sound too pessimistic, but I'm thinking that Bryan's challenge is going to have to focus as much on understanding and developing cultures of collaboration amongst teachers as it would scheduling and resource allocations.
Parting Thoughts
While it may have been personally beneficial for me to put a couple thousand words into a grand challenge I thought about on my own, I realize that our best hopes for meeting a grand challenge come when we share and push each other's ideas. As a student of curriculum and instruction, I find much to like in Robert and Patrick's thoughts about curriculum and David and Bryan's thoughts about instruction. There's some really meaty stuff there.
I've also tried to think about what
wasn't mentioned as a challenge. Nobody said, "I really think we need to better understand how students think about ratio/functions/number/proof/etc." While people are hard at work on such questions, I don't think there's any widespread perception that a lack of research in specific areas of student mathematical understanding is what is holding us back. (If there's a challenge I should be writing about, it's about the dissemination and use of this information.) I'm also happy to see that people weren't writing challenges involving new sets of academic standards. It's rather unfortunate that so much energy is being put into debating Common Core when it seems quite likely that standards account for little of the variability in student outcomes. We have a list of stuff we want students to learn. Fine. I'm ready to focus more of our efforts on the learning, not the list.
Lastly, to touch briefly on the challenge I hinted at near the top of this post, I didn't see any equity-focused grand challenges. I think I speak for Robert, Patrick, David, and Bryan when I say we all believe in achieving equitable participation and outcomes in mathematics education. Then again, we can't just say that and expect equity to come about by accident. There are elements of each challenge mentioned that could be used to promote equity, but it's going to take a more explicit focus than we've given it. In fact, maybe the first step is to significantly change the representation implied when I say "we." It seems simple enough, but privilege has a way of producing thoughts of "for" and "to" instead of "with," and that's a challenge for the kinds of people and organizations who pose challenges.
