2024 Year In Review

Teachers leading PD in the San Luis Valley

Where did the year go? In order to get at least one post up on this blog before the calendar flips to 2025, let me recap some big events of the last year.

January 2024: Math Routines in the San Luis Valley

The first highlight of the past year happened in Alamosa in January, and it was the result of many months of work. In Summer of 2023, my CDE colleagues and I recruited a dozen teachers from Colorado's San Luis Valley to attend conferences in Denver for four days and then turn what they learned into local professional development for districts in the San Luis Valley in January. I'm skipping over a lot of details, but it was great to work with these teachers and support them in their sessions.

February 2024: Launch Years Convening in Anaheim

It was a great year working with Colorado's Math Pathways Task Force and receiving support through the Launch Years Initiative. A contingent from Colorado joined up with more than 20 other states in a very rainy Anaheim, California to share progress and strategies about how to improve high school math and transitions for students as they matriculate to college and other postsecondary opportunities.

 

It rains in California, sometimes

July 2024: Math Intervention Design Workshop

In July, CDE teamed up with CCTM to host a design workshop in Summit County for a week. As I told the attendees, it probably felt like (a) work that felt like a vacation, or (b) a vacation that required a lot of work. Either way, Summit County in the summer is a delightful place to be. About 20 educators of all kinds -- higher ed, K-12, veterans, novices -- worked to develop materials that should be useful with students who are still struggling after regular instruction. We were all really pleased with the effort that went into these materials and we're looking forward to getting them into people's hands and getting feedback about how well they work.

The design workshop participants got to test their creations with summer school kids in Leadville, who graciously hosted us for a morning

September 2024: ASSM, NCSM, and NCTM in Chicago

This year's ASSM, NCSM, and NCTM conferences were in Chicago. To take in all three conferences requires a 9-day stay and by the time it's over, my head is swimming with all sorts of ideas and things I want to do and learn more about. ASSM held their conference in a hotel on Navy Pier, which was a treat.

In recent years, I've tried to blog at the end of each day of my conference trips. That didn't happen this year -- I was more social in the evenings than usual, with takes time, and I had two presentations at NCTM to finish preparing for. I kept thinking I'd get around to writing some recaps, but maybe I'll settle for a few bullet points here:

• A decade ago, I joked with many people that if you wanted people to show up at your session, all you needed to do was name it something like, "Common Core iPad Games for the Flipped Classroom." I don't know what would get the equivalent amount of attention today, but surely it has "AI" in the title. I've seen a few AI presentations so far and I'm still waiting for one that (a) focuses on the use of AI in math education specifically and (b) is practical without too much hype / too little skepticism.
• There's a national momentum for reforming high school math. I don't know exactly what the results of that momentum will bring, but I hope it's great. What we currently have isn't serving enough of our students as well as it should.
• I used to read stories about how many baby boomer teachers would be retiring. I think we're on the other side of that now, and it feels like we have a lot of younger and more novice teachers who can really benefit from the support they get at conferences and in their other professional learning. I know what teachers can get out of a 60- or 90-minute conference session is limited, but the overall exposure to new ideas and the enthusiasm behind them is something I hope every teacher has a chance to experience.

Navy Pier, Chicago, is a great place to enjoy a conference

October 2024: Fall Math Pathways Summit

To put an explanation point on Colorado's math pathways work, the task force hosted a summit in Colorado Springs. About 100 people attended from all over, and there's a lot of excitement about the work.

Thanks to Pikes Peak State College for providing a great space for the Fall Math Pathways Summit

Colorado Math Pathways Conference

On Monday, I attended the Colorado Math Pathways Conference, a gathering of faculty and department chairs from Colorado's 2- and 4-year colleges and universities, mostly representing mathematics departments, general education programs, and advising. In coordination with Colorado's Department of Higher Education, the Colorado Math Pathways Task Force has been examining the mathematics options required of Colorado's college students, and they've made these four recommendations:

1. Revise the curriculum to provide a calculus pathway, a statistics pathway, and a quantitative thinking pathway. Many students who would have previously taken college algebra would now take one of two courses: social and behavioral science majors would take statistics, and arts and humanities majors would take a course in quantitative reasoning.
2. Improve college advising through the creation of "meta-majors" with similar core and math requirements, and use multiple measures (beyond a sole placement score) to determine a student's readiness for college-level mathematics.
3. Support and develop instructors to teach pathway courses, especially the increased number of introductory statistics courses that will be needed.
4. Use multiple venues to improve communications about the pathways and their recommended use.

With these recommendations in mind, the purpose of the conference was to come together and work on some details of implementation. Colorado has thirteen 4-year institutions and fifteen more 2-year institutions, and a lot of effort over the years has gone into coordinating courses across the institutions to ensure transferability of general education courses.

Lieutenant Governor Joe Garcia led off the conference with comments about Colorado's successes in preparing undergraduates and our continued need to prepare more. Reflecting on a particular reform effort during his time leading Pikes Peak Community College, Garcia remembered a proposal to shift from three to four remedial courses prior to a gateway math course. When he looked at the data, he learned that fewer than 10% of students enrolled in remedial courses were actually making it to the gateway course. At Pikes Peak and elsewhere in Colorado, there has traditionally been more focus on enrollment and revenue generation than actually getting students through their programs. For an unfortunate number of students, they find themselves able to complete everything in their program except their mathematics requirement, and that is why the work of the Math Pathways Task Force is so important.

Dean Allison, the math department head at UNC, discussed the growing need for graduates to interpret and analyze data. For this reason, he and the task force want to join a national trend that moves away from college algebra as the default math course and towards courses in statistics and quantitative reasoning. This will require a lot of effort on the part of departments, advisors, and instructors.

The task force has been working with the Dana Center at the University of Texas at Austin, a national leader in mathematics education who has been working with multiple states on math pathway revisions. Uri Treisman keynoted the conference to describe some of the efforts, trends, and findings he's seen across the country. He said that math was currently under two different spotlights: the first, a harsh spotlight, that shows a lack of completion and high failure rates, and a second, a positive spotlight, where students with skills in mathematics are more successful negotiating rough job markets and becoming upwardly mobile in society.

Treisman continually stressed that the reality of today is different than the reality of 10 or 20 years ago. Right now, three times as many students take calculus in high school (almost 25% of all high schoolers) than in higher education, and those students tend to succeed at high rates in college. Studies of transcripts reveal that those succeeding in college calculus almost certainly took calculus in high school. Only about 1% of students going beyond college calculus to a higher level course (like differential equations) started college in college algebra. We've been placing students in college algebra to start them on a path towards math-intensive majors, but almost no students are actually following this path to get there. It's not that we haven't been trying to make this path work, either. At a meeting of community college leaders in Ohio, Treisman had the group list all the various initiatives for improvement that they've tried. They managed to list a total of 81 initiatives, most of which lasted 2-3 years, and zero of them were sustained long-term. People work hard, but too often people vital to the effort move elsewhere, money dries up, or 4-year institutions don't cooperate.

Uri Treisman and Amy Getz, comparing the success rates of a co-requisite support model vs. a pre-requisite support model.

Treisman recommended four coordinated innovations needed in new pathways:

1. Relevance: Too often college advisors accept students' wishes when it comes to course placement, rather than really trying to align the student's interests with the mathematical skills required of various majors. Placement tests haven't helped this, either. Said Treisman, "No longer should a placement test ensure a student is misplaced into a course, supported by a criminal enterprise of testing companies."
2. Acceleration: We've been trying to help students catch up by slowing them down in remedial courses, but we've learned that about two-thirds of students can succeed in regular courses when they're given proper support. There was a belief that these supports would ensure success for all students — a belief that was "polished by hope and unfettered by data" — but we now recognize that to see success for this other third of students it will take a national, Manhattan-type project to figure it out.
3. Narrow the gap between support and instruction: At Northwest Vista Community College in Texas, an advising "SWAT Team" crashes classes to confront students about why they're enrolled in the class they're in. About a quarter say it's because it fit their schedule, not because it's the class they need. They also aggressively supply support to students in the first three weeks of a course, before students slip behind or think there's a stigma about asking for help.
4. Introduce evidence-based strategies into instruction: Treisman says this may very well be the hardest part, yet it's less important relative to the structural improvements needed and described above.

Treisman then pointed out some key challenges. For one, math departments need to come to the realization that tradition shouldn't be the dominant force in program design. Math departments also need to push back against other programs, like business schools and psychology departments, who often use math prerequisites to weed students out of their programs. Math departments need to be a partner instead of a service discipline that is often placed as a barrier to student success. As Treisman said, "We no longer want math departments to be a burial ground for tens of thousands of people wishing to better their lives."

Amy Getz, also working with the Dana Center, spoke passionately about needed shifts in our thinking. We need to think beyond the classroom and past our old ways of unnecessary prerequisites, predictable outcomes, inconsistent course alignments, and a lack of clear messaging about math requirements.

The rest of the day was structured around breakout sessions and whole-group debriefs of those sessions. In the morning, the breakouts were organized around each of the quantitative reasoning, statistics, and calculus pathways. In the "StatsPath" breakout, I learned that the CU-Boulder math department is preparing to drop college algebra altogether, knowing that the easiest way to get advisors to stop recommending it would be to eliminate it. The discussion moved towards the details of the content of a prerequisite course students might take before statistics, but Uri redirected the conversation back to structural elements, which led to good conversations about transfer credits, the need for leadership, and developing the capacity to offer new courses. People in the quantitative reasoning breakout talked about how to work with partner disciplines, while the calculus path group reported on various ways to provide student support.

Afternoon breakouts centered on issues of advising, degrees with designation, and on-ramps for under-prepared students. With advising, we talked about the need to coordinate higher ed and high school advising, and to see if the meta-major model really helps advisors and students make decisions about the math courses that align with student interests. The degrees with designation group talked more about working with partner disciplines to determine the learning outcomes they want and why, and putting people in positions to advocate and create change in the system. The on-ramps group focused on issues of communication, including things like email lists, websites, blogs, virtual open houses, and coordinating with organizations like the Colorado Mathematics Association of Two Year Colleges (ColoMATYC), the Colorado Association for Developmental Education (CoADE), the Rocky Mountain Section of the MAA, and CCTM.

Feeling around in the dark to understand the elephant of local control

The view from my new office.

Educationally, Colorado is a "local control" state, and I know this because I've heard many people say it. I once read something that said "there are only six local control states," and I assumed Colorado was one of them. (I also assume many more than six states would claim to belong on the list.) The document did not list the states or describe the difference between local and not-quite-local. But by all other accounts I've seen and heard, Colorado is a local control state.

Today was my first day in my new job at the Colorado Department of Education, and at almost every turn I was reminded of Colorado's status as a local control state. Before I go further, let me be clear: I see many positives in local control and generally favor it. I appreciate that historically and politically we've entrusted our schools to the communities they serve, and I cherish my rural school experiences where the tight bonds between the school and the community generate a special sense of pride and responsibility.

Local control is not to be confused with insular, however. I've never known a school district where teachers and district officials didn't seek resources and guidance from outside their boundaries. In my new job, I'm now one of those outsiders who can offer some of those resources and guidance. But as an employee of the state, I have to be rather careful about the resources and guidance I give, lest anyone be confused about whether control lies with the state or with the school district.

It's a bit awkward, as you might expect. And it's becoming apparent to me how local control influences the things a CDE employee decides they can and cannot say is, well, a little arbitrary. Maybe arbitrary isn't the right description, but rather there's some socio-normative set of unwritten rules guiding all this that aren't 100% logical. For example, we believe local districts have the power to make curriculum decisions, not the state. But what does that mean? For me, I know without a doubt this means I have no power as a state employee to require that a district, school, or teacher use a particular set of textbooks. Okay, that's clear to me, but can I recommend a textbook, while plainly stating that textbook choices are not the state's decision? Again, this is something we expressly avoid at CDE because we don't want to give districts the impression that they're not in control of their curriculum. Let's make this even fuzzier: If someone asks me what textbooks I've personally used and why I chose to use them, can I answer? I certainly hope so, as I'd feel silly not answering the question. But with my answer, I have to be cautious that what I say is not construed as some kind of state endorsement of a particular textbook. Why? Local control, that's why.

Yet, there are numerous areas where we, the citizens of Colorado and our elected representatives, have decided the state should have control. The state dictates sets of academic standards, which are accompanied by all sorts of supporting documents. The state also requires the administration of assessments that measure student performance towards those standards, and by law the results of those assessments are used to hold local educators accountable. Whether I approve or disapprove of these things is irrelevant in the current discussion, because I only name them to illustrate how the state has control over things that might not be textbooks or curriculum, yet have undeniable influence over local educators' choice of curriculum and related materials. We still claim to have a system of local control, but it shows that local vs. state control of schools is beyond a simple binary classification.

I'm reminded of a pair of sessions I attended at the Realistic Mathematics Education Conference in 2013. The first was from several people from CDE, who presented a series of standards implementation materials they developed in cooperation with local educators. The second presentation came from national curriculum developers from The Netherlands, who showed their new integrated STEM curriculum materials. Both groups sincerely wanted high-quality educational experiences for students, but CDE illustrated the more traditionally American approach of building out documents and tools to support local decision-makers, instead of developing student-facing curriculum materials, as seen in The Netherlands. Dutch schools are not required to use materials developed by their national curriculum office, so you can say there's still an element of local control there. But at a national/state level they do experience a broader set of options for what they produce for schools, and these options are supported by a different political and cultural climate than we generally find in the United States.

Without belaboring the issue much further, I hope by now you get a little sense of the tension I'm feeling in my new position. I'm grappling with my new role and how my actions and words will be shaped by issues of local control, this elephant in the dark room that I felt my hands on all day. A lot of "what-ifs" came to mind, and some of them yield unsatisfactory possibilities. I'm frustrated by thoughts of continually almost doing curriculum work, and I don't think anyone wants to produce supplementary documentation just for the sake of producing it. With what I currently see offered for mathematics from CDE, there's a lot of value in the details, but it's a pretty overwhelming mass of stuff at first (and second) glance. That's not going to change overnight, but for me, after Day 1 on the job, I think the best thing I can do is think openly about this and to explore the boundaries of what is reasonable and possible. Ultimately, I serve the teachers and educators of Colorado, and for now, I can be honest about the thought I put into these things, even if I don't anticipate easy answers anytime soon.

New year, new job

Exciting news! Starting January 19, I will be the new mathematics content specialist for the Colorado Department of Education (CDE). Whenever people would ask me about what kinds of jobs I might be interested in doing after grad school, I always mentioned this one or ones like it. It's a mixture of curriculum work, policy, assessment, teacher education, and communication — all the things I've focused on during my time as a graduate student. Of course, there's only one such job in the state of Colorado, so it was my good fortune that the job came open (relatively) near the end of my time as a graduate student. To help ease the transition, CDE is generously allowing me to start the job part-time while I complete work here at CU-Boulder. Thankfully, my new colleagues at CDE are well-aware of the challenges of finishing a Ph.D. and are willing to be flexible so that I can complete my degree. As educators I think we're all on the same page when it comes to helping students (even a supposedly grown-up one, like me) see their program through to the end.

What will the new job mean for this blog and my other online activities? If I have anything to say about it, the blogging, tweeting, Google-plussing, etc. will only get better. (I may even have to venture more frequently into Facebook for my professional use, as well as LinkedIn.) As a researcher (even the graduate student kind), there's a large expectation to share with others in the research community. But in this role as math specialist, I plan to make sharing of research, policy perspectives, projects and collaborations, etc. part of my full-time work, with the goal of bringing it to math educators across Colorado and beyond. I look forward to sharing more of the great work I see teachers doing, along with the hard-earned lessons we collectively learn along the way. I also look forward to bringing more transparency to a state agency. Yes, there are a few things I can't say on the job (for example, in Colorado the state doesn't tell school districts what curriculum to use), but to the extent I can keep sharing, I will!

Monday Night at NCTM: Poster Session and Kenneth Zeichner's Plenary Address

Braving the snow, Ryan Grover and I slogged our way via RTD from Snowy Boulder to Almost-as-Snowy Denver. We arrived in time for a poster session. If you're unfamiliar with poster sessions, I'd describe them as a sort of academic mingling punctuated by awkward moments of silence as you read people's posters before deciding if the person is worth talking to. Yes, it's as fun as it sounds. I had one extended conversation with Sara Lohrman Hartman from Ohio University. Her recent dissertation work in rural Appalachia studied how a small school utilized a math coach. That's a rare luxury for a small rural school, at least in my limited experience.

As 7:00 neared I made my way to the plenary session and happily found Amanda Jansen standing outside the door. I'd never met Amanda in person, but I've conversed with her regularly on Google+ for more than a year. In that time she's made time for me and put up with my questions and rambling; it may not mean much to her, but as a grad student it means a lot to be welcomed by others in the field. We took some seats midway back in the room and settled in for a talk from Kenneth Zeichner, a well-known expert in the field of teacher education. His talk was not specific to mathematics education, but still of great interest for anyone who cares about how teachers are prepared and the institutions that prepare them.

My horrible picture of Ken Zeichner

As a student in Jennie Whitcomb and Dan Liston's Research on Teaching and Teacher Education class this semester, Ken's talk was very familiar. (If you've ever been told of the importance of reflective teaching, Ken and Dan literally wrote the book on the subject.) Ken began by framing three perspectives in the current debate about teacher education. First are the defenders, those who want to preserve the function and status of college and university-based teacher education (UBTE). Next are the reformers, those who seek to deregulate teaching and let the quality of teachers and teacher preparation be dictated by free market principles. Ken frequently cited Rick Hess as a proponent of this perspective, and shared George Will's belief that closing ed schools was the surest path to improving education in the U.S. Last are the transformers. This group, and where Ken places himself, wants reform of teacher education, but in a way that builds upon the capacity and expertise of UBTE and preserves a democratic, public system not driven by market forces.

From here, Ken shared what he sees as the two visions for preparing quality teachers. One is a system that produces professionals for a teaching profession; the other is a system that is technocratic and highly skill-driven, with teachers focused on actions that raise test scores in a narrow curriculum. Teaching skill is important, admits Zeichner, but professionals go beyond the management of a classroom to understand the social and political context of schools and their communities, and have skills that allow them to better reflect in ways that positively impact their teaching.

There are about 3.6 million teachers in the U.S., and alternative pathways for training those teachers (TFA, teacher residency programs, etc.) have grown since the early 1990s. Still, somewhere between 70-80% of the nation's teachers have earned licensure through UBTE at one of the almost 1400 college- and university-based programs around the country. Even if we were to deregulate teacher education, the market does not (and probably would not) have the capacity to prepare teachers without the help of UBTE.

Zeichner admits that UBTE has gotten a lot of non-critical, negative attention from the media. We've come to believe that too many teachers are not getting good results and efforts are now underway to trace poor student results back to the programs that prepared the teacher in those classrooms. When a recent bridge collapsed, said Zeichner, "The engineer who designed the bridge was exposed in the media but I don't remember anyone asking what engineering school he attended." Using another analogy, Zeichner said judging teacher education this way was similar to the idea that we should judge medical schools by patient outcomes. The result, says Zeichner, would be doctors avoiding treating those who need treatment most, just as teachers will avoid teaching in the tough schools with histories of poor test scores. This will lead to further inequality in how quality teachers are distributed, something the U.S. -- which was recently ranked 26th out of 29 nations in a UN report about child well-being -- cannot afford to have happen. The reformers' cries of "No Excuses!" does not rid us of the ails of poverty, no matter how good the teachers might be.

Zeichner addressed some key contradictions in the teacher education argument. For example, the Obama administration has pushed for higher standards for students while simultaneously pushing for alternative pathways to teaching that have lower standards. Also, as Mike Rose has pointed out, the logic of removing teacher education requirements and regulation, or to disregard teaching experience as a measure of quality, makes little to no sense in any other field. (Nobody would trust a neurosurgeon who hadn't been properly trained and lacked experience.) Further, no high-achieving country has deregulated their system of teacher education. Countries like Finland and South Korea have set their standards high, not lowered them.

Zeichner admits that messages like this, and his affiliation with schools of education, have led many to see him as a defender of UBTE. Zeichner insists that he's not a defender, but a transformer. Instead of deregulating UBTE, he sees a great need to change, yet not replace, our current system. This transformation would shift UBTE closer to the world of practice, providing a clinical experience for preservice teachers in a hybrid approach. This move would require the sharing of responsibility and accountability for teacher preparation with schools and their communities, a shared space with values that would prioritize democratized knowledge. Zeichner never advocated for this as a cure-all, but it is the approach most likely to support a system of effective, professionally prepared teachers.

Q&A

(Note: These quotes from Zeichner are rough approximations. Caveat emptor.)

Q: An attendee from Chicago asked about the dismantling of public education in Chicago and the closing of public schools in lieu of charters.

Zeichner: "I'm not against charter schools. I'm against disempowerment of parents and local stakeholders." Zeichner says he was seeing similar school closings in his home of Philladelphia, where public schools are being closed and replaced by charters run by charter management operators. A new study should be out soon that shows the ineffectiveness of these approaches so far. "There's a lack of democratic debate. I'm not for the alternatives to go away, but I'm for an honest debate."

Q: Why do educational entrepreneurs think they're going to make a lot of money?
A: Zeichner: "Some alternative routes use non-profit status for the tax breaks and they outsource their services. From a distance it's hard to see where they're making money, but they are. ... I'm not implying anything about morality. Greedy, self-serving attitudes are everywhere, including universities. But there is a lack of transparency about what's going on. It astounds me how the NewSchools Venture Fund has so much influence while flying under the radar."

Q: I've done UBTE and "quick fix" teacher education programs. Quick fix doesn't work, and online programs are dangerous. Still, too many teachers don't really their profession like a true profession.
A: Zeichner defends online education to a degree, having been involved with some online programs that offered a lot of quality. "I'd be careful in discounting all online programs. Some universities are using them along with more traditional approaches. They can be improved and we still have a lot to learn. We need to be careful about demonizing anything just based on their sponsorships or affiliations."

Q: What are things a new generation of mathematics educators should fight for in public education? How do we take our conversations outside the education community?
Zeichner: "For me, I've decided that ed schools alone will not accomplish much. The ownership of teacher education needs to be broader and shared with communities. I recently met with a lot of stakeholders and listened to their stories. There's a lot of community activism in places like Chicago and Philadelphia, and we can join those efforts already underway instead of coming in from above with solutions to their problems. I wish I'd been more in the public sphere in my career, writing more op-eds and participating in forums, even if many would have disagreed with me. Now, the polarization we're left with is harmful."

Q: How do we measure low-performing teacher education programs?
Zeichner: "I agree with the National Research Council's report on improving teacher education, and that the profession itself needs to take care of its own accreditation and monitoring. I'm afraid that test scores and value-added models will fill this role at great cost and distraction, as simply ranking as a way to improve quality has never really worked. To anoint these people who know little about the field as saviors is idiocy. Twenty-five years from now people will look back and wonder how we could have been so stupid."

(Mic drop. Not really, but I wish.)

My Thoughts

For me, Zeichner's serious (and some might say somewhat depressing) talk was balanced by a sense of validation. Not only did I recently write a paper that expressed many of these same ideas, it gives me hope that the system I'm a part of is recognizing a need and possibility for significant change. For some people, I'm sure a reformed/deregulated system of teacher education seems like a remote possibility, but right now in Colorado we're anticipating the introduction of legislation proposing to do exactly that, removing the requirement for teacher education as we know it. If that bill appears and makes progress, you'll be sure to hear more from me.

Planning for NCTM 2013

It's going to be a busy, busy week. The advantage of having a major conference in your backyard is that you feel like you can go to everything, but the disadvantage is the feeling you should be going to everything despite local distractions and responsibilities that don't go away. I try to get my money's worth at conferences; I think when I went to the Annual Meeting in 2008 in Salt Lake City I attended 17 sessions across the four days. We'll see if I set a new personal record this year. I'll be attending both the Research Presession and the Annual Meeting and have attempted to describe my schedule below.

RME at NCTM

Before getting into what I hope to attend, there are a number of opportunities to learn about Realistic Mathematics Education at this year's conference. I don't have a definitive list, but these are the RME-related sessions I'm aware of:

Tuesday

1:00 - Marja van den Heuvel-Panhuizen, Marc van Zanten, et al (Room 103, Colorado Convention Center)
Mathematics Curriculum Design and Development in the East and West

4:45 - +Fred Peck (Lobby A, Colorado Convention Center)
How Do Students Reinvent Their Mathematics? A Study Involving Slope

Thursday

2:45 - Mieke Abels, +Michael Matassa Jr., & +Raymond Johnson (Mile High 3 A, Convention Center)
Making Sense of Algebra with Realistic Mathematics Education

2:45 - +David Webb (Centennial Ballroom G/H, Hyatt Regency)
Promoting Student Reasoning and Understanding by Using Representational Pathways

(Putting Mieke and David head-to-head in the schedule was not something we RME folks wanted, but that's the way it is.)

Friday

12:30 - +Ryan Grover (702, Convention Center)
lim_{calculus → RME} f(calculus) = :-D

Saturday

10:00 - Britannica Digital Learning (Exhibitor Workshop, 304, Convention Center)
Meeting the Practice Standards with Models from Math in Context

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My Schedule

(Yes, I overschedule. It's good practice for when sessions are full, get cancelled, or when you just can't decide.)

Monday

5:30 - Jacqueline G Van Schooneveld (Lobby A, Colorado Convention Center)
Lesson Plan Evaluation Instrument: Assessing Math Lesson Plans

5:30 - Janet Mercado, Rossella Santagata, & Sonja Mohr (Lobby A, Colorado Convention Center)
Mathematics Knowledge and Beliefs and Their Relationships in Preservice Teachers

7:00 - Kenneth Zeichner (Rooms 205/207, Colorado Convention Center)
The Struggle for the Soul of Teaching and Teacher Education

Tuesday

8:30 - Anne Foegen, Barbara J. Dougherty, Vickie L. Spain, Jeannette R. Olson, and Subha Singamaneni (Room 107/109, Colorado Convention Center)
Building Progress Monitoring Measures for Algebra: Exploring Items and Scores

10:30 - I'll be planning for one of my presentations on Thursday, but if I weren't I'd probably be headed to Interactive Paper Session 9 (for papers here, here, and here) or Jere Confrey's session.

1:00 - Alison Castro Superfine, James Lynn, Timothy M. Stoelinga, Mara V. Martinez, Cynthia L. Schneider, and Diane J. Briars (Room 205/207, Colorado Convention Center)
Supporting Underprepared Algebra Students: Results from a Design-Based Research Program

1:00 - +Douglas Clements, Julie Sarama, Curtis Tatsuoka, Kikumi Tatsuoka, and Elvira Khasanova (Room 107/109, Colorado Convention Center)
Using Learning Trajectories to Create Cognitively Diagnostic Adaptive Assessments

3:00 - Dawn Berk, James Hiebert, +Amanda Jansen, Anne Morris, Laura Cline, Heather Gallivan, Erin Meikle, and Emily Miller (Room 102, Colorado Convention Center)
Effects of Mathematics Teacher Preparation on Teacher Knowledge and Practice

3:00 - +Pat Herbst, Daniel Chazan, +Karl Kosko, +wendy rose aaron, Justin Dimmel, Orly Buchbinder, and Ander W. Erickson (Room 104, Colorado Convention Center)

4:45 - +Fred Peck (Lobby A, Colorado Convention Center)
How Do Students Reinvent Their Mathematics? A Study Involving Slope

4:45 - +Edd Taylor and Tracey E. Dobie (Lobby A, Colorado Convention Center)
Religious Engagement and Context in Mathematical Problem Solving

4:45 - Corey Webel (Lobby A, Colorado Convention Center)
Secondary Mathematics Teachers Negotiating Obligations and Goals: Two Case Studies

4:45 - +Janine Remillard, Jacqueline G. Van Schooneveld, and Enakshi Bose (Lobby A, Colorado Convention Center)
Teacher Adaptations of Homework: A Window into Curriculum Enactment

4:45 - Sara Lohrman Hartman (Lobby A, Colorado Convention Center)
Rural School Math Coaching: Lessons from a Yearlong Case Study

Wednesday

8:30 - Geoffrey Saxe, Maryl Gearhart, Ronli Diakow, Nicole Leveille Buchanan, Jennifer Collett, Bona Kang, Kenton De Kirby, and Marie Le (Rooms 205/207, Colorado Convention Center)
Engagement in Mathematical Discussion: Linking Practices and Outcomes

10:30 - +Jo Boaler (Rooms 205/207, Colorado Convention Center)
Using Research to Make a Difference

1:00 - +Janine Remillard, Ok-Kyeong Kim, Luke Reinke, Napthalin A. Atanga, Joshua Taton, Dustin O. Smith, Hendrik Van Steenbrugge, and Shari Lewis (Room 105, Colorado Convention Center)
Using Curriculum Materials to Design and Enact instruction

3:00 - Interactive Paper Session 22 (papers here, here, and here, Room 110/112, Colorado Convention Center)

5:30 - Mayim Bialik (Bellco Theater, Convention Center)
The Power of Just One Teacher

Thursday

(If you can't find good sessions on Thursday, I'm pretty sure you're at the wrong conference. I wish I could be in multiple places at once!)

8:00 - +Douglas Clements and Julie Sarama (Capitol Ballroom 4, Hyatt Regency)
Math Lessons from Research

8:00 - Daniel Chazan, Mark Driscoll, and Megan Franke (Mile High 1 C/D, Convention Center)
Teaching and Learning of Algebraic Thinking: Research Insights

9:30 - Daniel Chazan (207, Convention Center)
Algebraic Thinking When Solving Equations and Doing Word Problems

10:30 - I'll be using this for last-minute planning for my afternoon sessions. If you think you might be interested in graduate studies in mathematics education, I'd recommend the 11:30 talk from +Amy Ellis, +Anita Wager, and Eric Knuth in 704/706 of the Convention Center.

12:30 - +Alan Schoenfeld (Four Seasons 2/3, Convention Center)
Meeting the Challenges of the Common Core Standards
(I'm sure I'll miss this one, but I hope it's on MathRecap.)

1:00 - +Raymond Johnson and +Susan Thomas (Mineral Hall D/E, Hyatt Regency)
Statistical Reasoning in the Middle School

2:45 - +Mieke Abels, +Michael Matassa Jr., and +Raymond Johnson (Mile High 3A, Convention Center)
Making Sense of Algebra with Realistic Mathematics Education

Friday

8:00 - Doug Sovde (Four Seasons 1, Convention Center)
PARCC: An Interactive Progress Update

9:30 - Steven Leinwand (Four Seasons 1, Convention Center)
Essential Mindsets for Tilling the Soil for the Common Core

11:00 - Emily Thrasher and Ayanna Franklyn (605, Convention Center)
Supporting Beginning Teachers through Online Social Communities

11:30 - Mark Ellis, Chris Stapel, and Cynthia A. Miller (704/706, Convention Center)
International Perspectives on Preparing Mathematics Teachers

12:30 - Kichoon Yang (Mile High 1 C/D, Convention Center)
NCTM Business Meeting

12:30 - +Ryan Grover (702, Convention Center)
lim_{calculus → RME} f(calculus) = :-D

2:00 - Matthew R. Larson (Mile High 1 C/D, Convention Center)
Implementing the Common Core State Standards: Five Paradigm Shifts

2:00 - Christian Hirsch (203, Convention Center)
Mathematical Modeling: The Core of the Common Core State Standards

2:45 - Heather Howell, Barbara Weren, and Shona Ruiz Diaz (110/112, Convention Center)
Mathematical Knowledge for Teaching Measures as Opportunities for Professional Learning

2:45 - NCTM Professional Development Services Committee (603, Convention Center)
Building Mathematics Learning Communities with NCTM Reflection Guides

3:30 - Francis Skip Fennell, John Wray, and Beth Kobett (Four Seasons 2/3, Convention Center)
Math Specialists Get Ready Now: Common Core Assessments Are Coming

Saturday

8:00 - Zalman Usiskin (Four Seasons 2/3, Convention Center)
Mathematical Understanding in the Common Core Standards

9:30 - Frank K. Lester (Mile High 1 C/D, Convention Center)
Whatever Happened to Problem Solving in the Math Curriculum?

9:45 - Cheryl Fricchione and Diane Austin (607, Convention Center)
An Invitation to Experience Online Lesson Study Firsthand

11:00 - +Dan Meyer (Four Seasons 1, Convention Center)
Tools and Technology for Modern Math Teaching

11:30 - Patrick Scott and Ann Lawrence (607, Convention Center)
Lessons from the U.S. National Presentation at ICME-12 in Korea

12:30 - Vi Hart and George Hart (Four Seasons 2/3, Convention Center)
Viral Math Videos: A Hart-to-Hart Conversation

Reminder: You can catch some of what you might miss at MathRecap, and if you're attending the conference I encourage you to contribute!

What Makes a Student Public? An Alternative Outcome for the Douglas County Voucher Program

Although the ruling came in about two weeks ago, lately the Douglas County voucher program hasn't been far from my mind. I credit George Will's column in Friday's Washington Post for making me rethink the case and its outcome, and finally motivating me to organize some loose thoughts that have been floating around in my head.

If you aren't familiar with the case, it basically boils down to this: The Douglas County School Board believed so strongly in school choice that it voted to give 500 "choice scholarships" (vouchers) to its own students to attend area private schools. The scholarships are worth $4,575, or 75% of the district's per pupil revenue. (The district is keeping the remaining 25%.) Several groups representing the interests of taxpayers and those worried about public funding of private religious schools filed lawsuits, and on August 12th Judge Michael Martinez ruled in their favor, saying the plan "violates both financial and religious provisions" of the Colorado Constitution. Some of the consequences of this ruling are unclear, as many students have already accepted part of their scholarships and are enrolled at private schools.

I was a bit surprised by Judge Martinez's ruling. Not because it came down on the side of the plaintiffs, but because it used the public funding of religious schools as a primary reason. In reaching this decision, Martinez seems at odds with Zelman v. Simmons-Harris, a 2002 case where the Supreme Court upheld an Ohio voucher program primarily because the vouchers went to parents and not directly to religious schools. Although I don't know many of the details surrounding either Zelman v. Simmons-Harris or the DougCo case, George Will's assertion that the two are "legally indistinguishable" seems to have some merit. But that got me thinking -- what if Martinez had found different reasoning for his decision, one not relying on religion at all?

I think we tend to focus the debate on public vs. private schools. Instead, let's focus on students. Previous Supreme Court decisions have upheld both students' rights to receive a free appropriate public education and attend private schools, as well as receive vouchers. Students who attend public schools are public school students. Students who attend private schools (without vouchers) are private school students. But what are students who use vouchers to attend private schools? Public or private? Can we have something in-between?  If so, what rights do those students have?

In Douglas County, students who receive the voucher are still required to take the CSAP, Colorado's annual standardized test. Participating private schools are required to provide information to the district about their attendance and the qualifications of their teachers, as well as be willing to waive requirements that participating students attend religious services. It's clear that these provisions are included to meet various requirements of federal education law, namely No Child Left Behind's testing and "highly qualified educator" requirements. These are requirements of public schools and public school students. And let's not forget that Douglas County is keeping 25% of each student's share of state funding. Could it do this if it didn't claim the students were not, at least in some way, students of the Douglas County School District?

Instead of essentially upholding the Establishment Clause, what if Judge Martinez had instead declared the DougCo "scholarship" students as public students and decided that voucher students had the right to simultaneously receive both a free and private education? In essence, what if he had told Douglas County that the vouchers would be legal so long as they covered the entire cost of each student's education at their chosen private school? If the court decided that (a) by benefiting from public monies, the students were public school students (and there is no murky in-between), and (b) public school students have the right to a free education, then (ironically!) Douglas County would be facing a difficult choice about whether or not a voucher program was in their best interest. As proponents of school choice it would be awkward for the district to back away because they couldn't afford the vouchers, although the high cost is surely what keeps many families away from private schools, voucher or no voucher.

At this point I realize that my knowledge of the law is rather limited and this issue was probably dealt with a long time ago. Still, I find it an interesting perspective and it makes me want to hunt down Kevin Welner (who knows a thing or two about vouchers) in the hallways next week to ask him about it. If any of you have any knowledge or thoughts you'd like to share, I'd love to hear it in the comments below.

Planning for the 2010 NCTM Regional Conference in Denver

A week from now I'll be exhausted and headed home after a couple of long days at the Denver Convention Center and the NCTM Regional Conference. I've been planning my schedule and this is what I have so far:

Thursday, October 7:
9:30 - 10:30 "Learn <-> Reflect Kickoff: Geometry Must Change! Enough Trivia!" with Johnny W. Lott (Past President, NCTM; University of Mississippi) OR "A Learning Trajectory for Fractions with Low-Achieving Students" with Maria Ables (Freudenthal Institute)
I have class Thursday morning so I'll probably miss both of these. The content of Lott's presentation sounds good and pretty much everything from the Freudenthal Institute is good.

11:00 - 12:00 "Research-Based Practices that Increase Student Achievement -- and Practical Suggestions for Implementing Them In Your Classroom" with Diane Briars (President, National Council of Supervisors of Mathematics)

12:30 - 1:30 "Beyond Common Core Standards: Deep Curriculum Needed to Link Instruction, Assessment, and Professional Development" with Lorrie Shepard (University of Colorado at Boulder) OR "From Informal to Formal with a Click: Using Technology to Facilitate Progressive Formalization in Algebra" with Fred Peck and Jenn Moeller (Boulder Valley School District)
I had class with Fred and Jenn last summer and would love to see their presentation, but I really want to see Dr. Shepard's take on the Common Core Standards and the curriculum we need to support it. In either case, I might have trouble getting to Denver in time.

2:00 - 3:00 "Students' Geometric Reasoning: Research, Assessment, Teaching, and Intervention" with Michael Battista (Ohio State University)

5:00 - 6:30 "Why Videogames Are the Perfect Medium to Learn School Mathematics" with Keith Devlin (Stanford University)

Friday, October 8:
8:00-9:00 "Engaging Middle School Students in Proving" with Eric Knuth (University of Wisconsin -- Madison) OR "Online Professional Development Collaborative for Grades 6-12 Mathematics Teachers" with Robert Mayes (University of Wyoming) OR "Interactive, Visual Tools to Support the Learning of Algebra" with Henk van der Kooj (Freudenthal Institute)
I've seen probably seen the visual tools Henk van der Kooj will show at previous Freudenthal events, but if you haven't seen them you should check them out.

9:30-10:30 "NCTM and Implementation of the Common Core State Standards" with David Masunaga (Board of Directors, NCTM) and Michael Shaughnessy (President, NCTM)

11:00 - 12:00 "Students Reasoning and Sense Making on Some Favorite Geometry Problems" with Michael Shaughnessy (President, NCTM)

12:30 - 2:00 "Accessible Assessment: Selecting and Designing Tasks That Show What Students Know" with David C. Webb (University of Colorado at Boulder)
I've taken a class on assessment with Dr. Webb, so I may look for something else I haven't seen.

2:00 - 3:00 "Classroom Conversations: The Heart of Teaching" with Gail Burrill (Past President, NCTM; Michigan State University)

I still have some gaps to fill. I'll spend some time in the exhibitor hall, confirming my (somewhat baseless) distaste of most for-profit publishers and seeking out friendly faces. Hope to see you there!

A Week for Irony and Contradictions in Education

The recent past has been wrought with irony (or at least contradictions) in all corners of education. Let's look at a list:

Read Bruce Baker's "If money doesn't matter..."  Bruce examines the argument that "we keep throwing money at education and it hasn't made a difference," and points out that (a) schools with lots of money tend to do well, and (b) people who make that argument don't mind throwing money at charter schools.

Just days after Vanderbilt University released their study finding no improvement in test scores in Nashville's merit pay system, schools around the country (including Colorado) received millions of dollars from the federal government to implement merit pay systems.

NBC's "Education Nation" summit will gather "the foremost policymakers, elected officials, thought leaders, educators, members of the business community and engaged citizens" to discuss issues in U.S. education. Unfortunately, the list of invited panelists NBC is promoting doesn't include any teachers, students, principals, or professors. The only university-affiliated participant on the list is the President of the University of Phoenix, who happens to be a major sponsor of the event.

In a post called "Does Education Pay?" the Center for College Affordability and Productivity (CCAP) criticize the College Board for "a confusion of correlation with causation." The day before, the CCAP asked, "Should We Abolish Colleges of Education?" and use this logic:

1. U.S. students "perform in a mediocre fashion" on international tests.
2. Kids need remediation and/or drop out of college because of their mediocre education.
3. Good teaching is better than mediocre teaching.
4. Most teachers studied at a college of education.
5. The teachers who didn't go to a college of education are as good or better than those who did, such as Teach for America teachers. (Sorry, CCAP, that's rarely true.)
6. Colleges of education support anti-knowledge and anti-intellectual biases and make their poor students look good by inflating grades.
7. Colleges of education don't want teachers to be rewarded for student learning because student self-esteem is more important than knowledge.
8. While there might be some good colleges of education, most of the people who really understand education are not in education schools.
9. Courses in education are less helpful for math teachers (for example) than advanced math courses. (This was not the finding by Floden and Meniketti (2005), who say it's not as simple as "more math is better.")
10. THEREFORE, we should close colleges of education, which are a "blight on true 'higher education' [that] should be discouraged at all institutions depending on taxpayer funds."

I admit, that post had no trouble working its way under my skin. If you can keep track of all the assumptions and correlation/causation confusions in their argument, you're doing better than me.

If I missed anything you want to add to this week's list, feel free to add them in the comments.

References
Floden, R. E., & Meniketti, M. (2005). Research on the effects of coursework in the arts and sciences and in the foundations of education. In M. Cochran-Smith & K.M. Zeichner (Eds.), Studying teacher education: The Report of AERA Panel on Research and Teacher Education (pp. 261-308). Mahwah, NJ: Erlbaum.

Colorado's Amendment 61: The Simple Facts

Several stories have hit the news recently about Amendment 61, which Coloradoans will vote on this November. Here's what you need to know:

• The property taxes that Colorado school districts rely upon for their funding usually start coming in during March.
• When Colorado school districts used to budget around a calendar year (Jan-Dec), they didn't have long to wait (~3 months) for property tax funds to arrive.
• When Colorado school districts switched to fiscal years (Jul-Jun), they had to wait much longer (~9 months) for property tax funds. Since districts don't have an extra 6 months of cash lying around, the state has provided interest-free loans to school districts to bridge the gap.
• A provision of Amendment 61, if passed, prohibits the state from making these loans to school districts, and many districts aren't sure what they will do (or, better put, not do) without that money.

I don't think I need to tell you how to vote, but this does sound like an Amendment worth studying carefully. Ed News Colorado has the best coverage I've seen yet, including links to sites for and against Amendment 61.

We're Not All Math and English Teachers

Yesterday the Des Moines Register published an editorial applauding Colorado for reforming its teacher evaluation laws. The editorial goes on to criticize Iowa's reforms, saying the state is "moving too cautiously." Iowa requires teachers to "provide multiple forms of evidence of student learning and growth," but the Register is disappointed that the inclusion of standardized test scores is not required.

When will the public and policymakers realize that not every subject is covered by a standardized test? I feel like this is one of the most overlooked aspects of this argument. We're not all math and English teachers! Colorado spends 3 hours, per subject, each school year to measure every grade 3-10 student's achievement in math, reading, and writing (plus 3 hours for science in grades 5, 8, and 10). In a state that requires a minimum of 1080 hours of student contact time, this minimum of 9 hours of testing represents less than 1% of a school year. If only it felt like so little!

But that's only for math and English. If you want to mandate teacher evaluations based on standardized tests, you need equally rigorous tests for every subject and every teacher. Can you imagine tests for P.E., music, art, or vocational courses? What if schools could only offer classes that were backed by standardized tests?

Let's also consider the extra time required. Suppose students average 7 Carnegie units (credits) per year. At 3 hours per subject and 7 subjects, we're up to 21 hours of testing per year. If you were to require that standardized tests measure student growth, you'd need to test each student both at the beginning of the year and at the end. Doubling the testing takes us to 42 hours, or almost 4% of the year. It still may not sound like much, but students aren't going to be testing 7 hours a day for 6 straight days. If students tested 2 hours daily, the testing schedule would stretch out to 21 days, or about a month of the school year (half at the beginning, half at the end). You thought finals week was bad? Try two weeks!

Unfortunately, I have yet to work in any school that could carry on with regular learning during standardized test times. We tried everything from one test per day to four, mixed with full-length classes meeting on a rotation to all classes meeting on a shortened schedules. No matter how little testing is done, that testing time affects everything else in the day. Can you imagine a month spent like that?

This little rant has come from me, a guy who has almost always been pro-test. They have their place in the educational system and are part of an effective assessment strategy. But when people want every teacher to be measured by their students' standardized test scores, we have to think about the possible ramifications. So be careful what you wish for, Des Moines Register!

Functions of Functions

Mathematical functions are usually introduced formally to students somewhere around the end of Algebra 1 or maybe in Algebra 2. If my Algebra 2 final exam had included the question, "What do you know about functions?" I probably would have said: (a) You use \( f(x) \) instead of \( y \), and (b) if a graph fails the vertical line test, it's not a function. With all due respect to my high school math teacher, this would have been a lousy answer. I might have known some peripheral information, but not the core understanding. Sadly, the mathematical importance of functions is not that difficult conceptually, yet it's crucial to the majority of the content learned in Algebra 1 and 2. Yet students still struggle with what makes a function a function.

My first experience trying to teach the definition in a non-traditional way was using the "Cola Machine" problem in CPM Algebra (Math 1). The problem, several days into Unit 11, describes the following:

The cola machine at your school offers several types of soda. Your favorite drink, Blast!, has two buttons dedicated to it, while the other drinks (Slurp, Lemon Twister, and Diet Slurp) each have one button.

1. Explain how the soda machine is a relation.
2. Describe the domain and range of this soda machine.
3. While buying a soda, Mr. Hagen pushed the button for Lemon Twister and got a can of Lemon Twister. Later he went back to the same machine but this time pushing the Lemon Twister button got him a can of Blast! Is the machine functioning consistently? Why or why not?
4. When Karen pushed the top button for Blast! She received a can of Blast! Her friend, Miguel, decided to be different and pushed the second button for Blast! He, too, received a can of Blast! Is the machine functioning consistently? Why or why not?
5. When Loufti pushed a button for Slurp, he received a can of Lemon Twister! Later, Tayeisha also pushed the Slurp button and received a can of Lemon Twister. Still later, Tayeisha noticed that everyone else who pushed the Slurp button received a Lemon Twister. Is the machine functioning consistently? Explain why or why not. (Sallee, et. al., 2002, p. 375)

Most textbooks define functions approximately the same way: a relation is a function if there exists no more than one output for each input. Without a context, however, that definition might not carry much meaning, and relying solely on the vertical line test in a graph may not be helpful enough for many students. The soda machine problem, with its emphasis on consistency, gives both teacher and student a very approachable context within which to discuss what a function is and is not.

Early in my teaching career I thought Algebra 1 basically boiled down to two big ideas: solving equations and graphing lines. If students could do those two things, I felt pretty good about them passing my class. Now I see the big ideas of Algebra 1 differently and basically aligned with Colorado's revised standards for high school mathematics. The six expectations listed for Colorado's high school algebra standard can be summarized as follows:

1. Functional representations (equations, graphs, and tables)
2. Function behavior (qualitative)
3. Function transformations (parameters and parent graphs)
4. Equivalent expressions, equations, and inequalities
5. Solving equations, inequalities, and systems of equations
6. Mathematical modeling using functions

Four of the six expectations explicitly mention functions. The last expectation, mathematical modeling using functions, represents (for many math educators) the ultimate goal for math instruction: to give students the mathematical power to describe and understand the world around them. Instead of just solving and graphing, the big idea of high school algebra is functions, with most linear function work in Algebra 1 and most non-linear function work in Algebra 2.

I think it's a mistake to delay the explanation and definition of functions until late in Algebra 1 or later. Algebra 1 and younger students can understand the cola machine problem or other, similar contexts. Suppose the class plays an "exchange" game. Student A gives the teacher three triangles in exchange for two squares. What should Student B expect to get in exchange for his three triangles? What exchange would represent a function versus a non-function? For something based more in the real-world, this conversation could be set in the context of currency exchange. Also, a helpful model for learning functions might be function machines, such as:

Function machines are helpful models for learning functions, function composition, inverse functions, and even solving equations. They help stress the input-output relationship in a way that words or equations alone might not. Students will also naturally expect a single output for each input. The real challenge for Algebra 1 or younger students is to present a variety of equations that aren't functions, or else risk having students think that every two-variable equation describes a function.

Patterns can also be used to teach functions. If students are given the sequence 2, 4, 8, …, some students are likely to predict 14 as the next term (adding 8 plus 6, the next consecutive even number), while other students might predict 16 (the fourth power of two). Because the fourth term could reasonably be two different values, we can't establish a functional relationship to describe the sequence. This could even be an example worth graphing to discuss the meaning of the vertical line test:

Our calculators also help us distinguish functions from non-functions. If you try to graph a circle on a Texas Instruments graphing calculator, you have to enter two functions: one for the top half of the circle, and one for the bottom half. Therefore, the written forms of the equation of a circle (such as \( x^2 + y^2 = 1\) or \(y= \pm \sqrt{1-x^2} \)) can't be functions. The graphing calculator is also a good tool for discussing the square root function, and why its graph must only be half of a parabola's inverse if we want it to be a function.

There two key obstacles that are likely to remain in a student's way of understanding functions. First, students will continue to focus the vast majority of the time and effort on functions, and there are too few real-world examples of useful non-functions to help distinguish the two. Non-functions are less powerful and less common, but without them we risk having students who casually accept that any equations with an input-output relationship is a function. The second obstacle is notation, and it's not an obstacle we can likely avoid. The change to function notation, such as using \(f(x) = 3x - 2\) instead of \(y = 3x - 2\), is not only difficult to explain as something other than an arbitrary change in symbols, but includes two aspects that are directly contrary to a student's prior knowledge. Now a letter (such as the \(f\) in the preceding example) is no longer a variable, but a name with no numerical value by itself. Function notation also uses parentheses to represent something other than multiplication, adding more work to the already overloaded duties mathematicians place on these two simple arcs. The power of function notation is the preservation of the input (\(x\)) with the output, but it's confusing that the output (\(f(x)\)) reuses the input variable, uses a letter that isn't a variable, and uses symbolism for multiplication of variables that's no longer multiplication. Perhaps a question like this could help students realize the power of the notation:

1. If \(y=16\), \(y=49\), and \(y=64\), what might be an equation relating an input, \(x\), to the output, \(y\)?
2. If \(f(4)=16\), \(f(7)=49\), and \(f(-8)=64\), what is \(f(x)\)?
3. Are the two questions above the same? Which one is easiest to understand? Why?

References
Sallee, T., Kysh, J., Kasimatis, E., & Hoey, B. (2002). College Preparatory Mathematics 1 (Algebra 1). (L. Dietiker, Ed.) (2nd ed., Vols. 1-2, Vol. 2). Sacramento, CA: CPM Educational Program.

Survey: Teaching to the Test

Colorado math teachers: I need your input! I have heard many teachers use the phrase "teaching to the test" and I want to know two things:

1. What does "teaching to the test" mean to you?
2. How well do you know the math CSAP test? What is assessed most? And least?

To help answer these questions I've developed a series of surveys, one for each grade-level math CSAP. To take the survey for your grade, please follow the appropriate link below:

• 3rd Grade Math CSAP
• 4th Grade Math CSAP
• 5th Grade Math CSAP
• 6th Grade Math CSAP
• 7th Grade Math CSAP
• 8th Grade Math CSAP
• 9th Grade Math CSAP
• 10th Grade Math CSAP

I need your responses by Monday, April 26. If you teach multiple grade levels of math, you're welcome to complete the survey for multiple grades. (Just be consistent with your name and email, please!) If you aren't a Colorado math teacher but happen to know one, please pass this post along to them so they can complete the survey,

Many thanks for your cooperation, and I'll be sure to let everyone know the survey results and how they factor into my research. If any of you submit perfect rankings, I believe you deserve some recognition. I'd offer cash prizes, but that's not exactly in my budget. Sorry!

Calling Colorado Teachers: Can I Analyze Your Assessment?

Sometime in the next week or so I need to observe a math or science teacher. I'm turning to you, my PLN, for a volunteer. I'm open to any grade level, and here's the official description of the assignment:

Analysis of the Assessment Practices of a Math/Science Teacher
For this assignment you will observe a school mathematics or science teacher to analyze opportunities to assess student understanding embedded within a classroom context. Determine the extent to which assessment is embedded in instruction, detailing the kinds of questions and tasks used during instruction.

1. Consider what it means to assess understanding
2. As part of your observation, be prepared to distinguish between questions that guide the discussion and questions that elicit student understanding
3. To what extent does the teacher create opportunities to gauge student thinking and learning? When does the teacher "check in" with students?
4. To what extent do students responses influence instruction?

Additionally, as part of this assignment include a brief interview of the teacher’s beliefs about assessment. Some questions to consider…

1. What is the purpose of assessment?
2. When do you assess your students?
3. [if appropriate] How do you use information from observations and listening to students to inform instruction?

These are just suggested questions. You are encouraged to adapt these and include other questions that you consider important. The report can be set up as a 2-part narrative that summarizes the observation and the teacher’s responses to the interview. Be sure to highlight specific aspects of what was observed and discussed, so that you can report your findings to the group and class on April 13th.

To be clear, I am not wishing to observe you giving a test or quiz, or to analyze the content and design of your test or quiz. I want to observe you during an instructional task and analyze what questions you ask, how you ask them, and how your students respond. As you can see from the assignment, I'm looking for the formative processes you use to determine if your students truly understand the content.

So if you don't mind an extra body in your classroom and can spare some minutes to discuss assessment, please email me at johnson@downclimb.com or DM @MathEdnet on Twitter. If you don't feel right for the study but can arrange something with a teacher that you think would be great to observe, that would be great, too. I'm in Boulder, am willing to travel a reasonable distance, and am available any day of the week. Your help will be greatly appreciated and I look forward to working with you!

Trying to Understand Colorado's School Funding Problems

Yesterday Colorado's Senate Education Committee approved House Bill 10-1369, the school finance bill. If passed, it requires every school district to reduce the state share of their total program funding by 6.35 percent. (State Share = Total Program - Local Share) But here's the catch: seven Colorado districts (Clear Creek, West Grand, Gunnison, Estes Park, Park, Aspen, and Summit) collect less than 6.35 percent's worth of state aid. For example, Park only collects $81 of state aid. (This seems impossible, but that's what the state says.) Instead of losing just that $81, they have to reduce local revenues (which means reducing their override mil levy) by more than a quarter million dollars so they have the same 6.35 percent reduction as every other district. How cutting local revenues helps the state budget is beyond me, so it seems the legislature is doing this in the name of equity. I'd like to think I'm somehow reading the information incorrectly, but I'm afraid I'm not.

(Update 3/31/2010: The Colorado State Senate approved the school finance bill, but thanks to an amendment proposed by Bob Bacon of Fort Collins, the seven districts named above will not suffer any loss of local funding.)
(Update 4/16/2010: After the House rejected Bacon's amendment, a committee agreed that the seven districts will lose money through state categorical grants. I'm not totally sure what that money is, but I hope to find out. The compromised bill still has to be approved by both houses.)
(Update to the update 4/16/2010: Previously I had confused the "factors" from the "categoricals," and the post now recognizes them properly.)

It may sound crazy, but it's just another example of Colorado's school funding problems. Colorado's school finance formula, established in 1994, determines base funding (a set per-pupil amount) and funding factors (extra money for cost of living, personnel costs, district size, at-risk students, and online students). Together, these define "Total Program" funding. In addition to Total Program funding, some schools get state "categorical" funding, which provide extra money for six categories: small attendance centers, English language proficiency, gifted and talented, special education, transportation, and vocational education. Colorado's tax revenue and education spending picture is defined by three sometimes contradictory laws: the Gallagher Amendment, TABOR, and Amendment 23.

Gallagher, passed in 1982, protects Coloradoans from increasing residential property taxes. Unfortunately, it might be doing that job too well -- Gallagher mandates that residential properties account for 55 percent of all property tax, the same as in 1985. The housing boom of the 1990s and early 2000s added greatly to the value of Colorado's residential property, forcing property tax rates down significantly in order to keep revenues in line with the 55 percent rule.

TABOR (the Taxpayer's Bill of Rights), passed in 1992, puts revenue and spending limitations on all levels of Colorado government, including schools. Government can't grow any faster than inflation plus population growth. In good years, excess revenue is returned to taxpayers. Unfortunately, that means Colorado has a difficult time building up a rainy-day fund in the event of hard times, such as those we have now. In 2005 Coloradoans passed Referendum C, a five-year "time-out" from TABOR, but during that time the state didn't have many opportunities to stockpile surpluses.

The restrictions of Gallagher and TABOR were causing some pretty dire restrictions on education funding, so in 2000 voters passed Amendment 23. The amendment was to restore funding to pre-TABOR levels and mandate minimum increases (at least the rate of inflation) for education spending. You should be wondering: if Amendment 23 requires spending increases, how is the legislature decreasing school spending next year? Due to the tight budget, lawmakers have had to "reinterpret" Amendment 23 to apply it only to base per-pupil funding, and not categorical factors.

So what are the consequences? Gallagher lowered property taxes, reducing one of a school district's primary local revenue sources, and pushed greater funding responsibility onto the state's general fund. TABOR placed restrictions on the general fund, causing underfunding of schools. Amendment 23 promised better funding for schools, but it didn't solve the revenue problems and Colorado still ranks very low nationally in various measures of K-12 spending. There's a current movement to exempt education spending from TABOR, but doing so would allow the legislature to backfill spending in other areas with education dollars, then raise taxes to fill the education hole. Whether you like TABOR or not, creating a loophole is probably not the best way to untangle Colorado's education spending knot.

One effort to untie the Gallagher/TABOR knot is the Lobato v. State case. That case challenges the state's constitutional responsibility to maintain a "thorough and uniform system of free public schools." If successful, the lawsuit would force Colorado to re-examine its 1994 school finance formulas, which in turn could force changes in Gallagher and TABOR. Defining "adequate" school funding isn't easy, and much of what was done in 1994 involved the political negotiation of available funds, and not an analysis of how much is actually costs to educate a student. There are three common ways to do this:

1. The "Successful School District" Approach (Empirical Identification) - Identify school districts that are performing at or above a set level, then examine how much it's costing them
2. The "Professional Judgement" Approach (Resource Cost Model) - Ask experts to estimate the resources necessary for success
3. The "Statistical Prediction" Approach (Cost Functions) - Use statistics to extrapolate funding based on extrapolations of current data

A good adequacy analysis will use multiple approaches and look for agreement between the models. Analyses recently completed for Colorado suggest adequacy levels near 150 percent of current state funding. Such increases could dramatically change education and school funding in Colorado for the long-term. In the short-term, however, it appears there is no fix to the current budget hole.

(If I've been incorrect in any of the above, by all means, suggest a correction in the comments!)

Boulder Valley Students "Earn" Zeros on Mis-administered CSAP

According to a story in Boulder's Daily Camera this afternoon:

"Sixty-seven students at Superior's Eldorado K-8 School will receive zeros on the writing portion of this year's CSAP test after an eighth-grade teacher broke the rules by having students practice on writing topics copied from a previous year's test."

Two years ago, the exact same situation happened at Runyon Elementary in Littleton. In both cases, the schools realized what happened soon after the test was administered and contacted CDE to report their concerns. In both cases, it appears CDE will give a score of zero to every affected student. This situation upset lawmakers enough in 2008 to include a provision for it in their CAP4K, more officially known as Senate Bill 08-212. It amended Colorado Revised Statute 22-7-604(3) to read:

"...the department shall identify and implement alterations in the calculation method, or other appropriate measures, to ensure that, to the fullest extent practicable, a public school is not penalized in the calculation of the school's CSAP-area standardized, weighted total score by inadvertent errors committed in the administration of an assessment."

So why doesn't that law apply to this new case in Boulder? Simple: it was repealed last year by Senate Bill 09-163, for reasons I never heard discussed. Assuming this is as honest of a mistake as BVSD claims, it doesn't seem fair to give kids zeros. After all, I'm sure the kids know more than nothing. I hope we hear an explanation from CDE and the Colorado legislature about the repeal of the CAP4K provision, and I won't be surprised to find it back in the law soon.