RYSK: Cobb, Zhao, & Visnovska's Learning From and Adapting the Theory of Realistic Mathematics Education (2008)

This is the 21st in a series describing "Research You Should Know" (RYSK).

It's Open Access Week (#OAweek) so I thought it would be fitting to use this "research you should know" post to highlight one of my favorite open access articles in mathematics education, Learning From and Adapting the Theory of Realistic Mathematics Education by Paul Cobb, Qing Zhao, and Jana Visnovska. Because the article is open access, I get to be less interested in summarizing it and more interested in giving you a reason to read it.

Realistic Mathematics Education (RME) is a theory for the design and development of mathematics curriculum. It is still deeply rooted in the Netherlands, where Hans Freudenthal greatly influenced mathematics instruction there with his belief that mathematics was a human activity, and that activity was characterized by mathematizing the real or readily imagined world. ("Realistic" comes from the Dutch phrase "zich realiseren," which in English means "to imagine.") This mathematization can be thought of in two ways, horizontal and vertical: "horizontal mathematization involves going from the world of life into the world of symbols, while vertical mathematization means moving within the world of symbols" (Freudenthal, 1991). Hans Freudenthal died in 1990 but his work continues, primarily at the Freudenthal Institute for Science and Mathematics Education at the University of Utrecht in the Netherlands.

There have been four primary avenues where RME has established itself in the United States. The first is with the middle school curriculum series Mathematics in Context, which grew from a partnership between mathematics education researchers at the University of Wisconsin (primarily Thomas Romberg) and researchers at the Freudenthal Institute. The second is the K-8-focused work of Mathematics in the City, which primarily brought together Cathy Fosnot from the City College of New York and Maarten Dolk of the Freudenthal Institute. The pair also wrote most of the Young Mathematicians at Work book series. The third place where RME is established in the U.S. is here at CU-Boulder, home of Freudenthal Institute US and its director, David Webb. David worked on the Mathematics in Context project at Wisconsin, and brought FI-US with him to CU-Boulder. The fourth place I recognize RME having a significant influence in the United States is in the work of Paul Cobb, particularly in his long research partnership with Koeno Gravemeijer, a researcher from the Freudenthal Institute. Cobb and Gravemeijer spent more than a decade working and publishing together, and that work did a lot to strengthen ties between RME as a design theory and theories in the learning sciences.

Like any idea or theory, RME has limitations. Over its 40+ years of existence it's proven to not be a static thing (van den Heuvel-Panhuizen, 2002), and this article by Cobb, Zhao, & Visnovska describes some of the important ways their work has both informed and been influenced by RME. They describe three adaptations: the first involves accounting for classroom activity and discourse in RME, the second acknowledges the mediating role of the teacher in making curriculum modifications and adaptations, and the third looks at how RME can focus on teacher learning, not just student learning. For details, I'll let you read the article for yourself at http://educationdidactique.revues.org/276. If you have any questions about the article or RME, leave a comment, find me on social media, or email me. We RME folks want to spread the word!

References

Freudenthal, H. (1991). Revisiting Mathematics Education: China Lectures. Dordrecht: Kluwer.

van den Heuvel-Panhuizen, M. (2002). Realistic Mathematics Education as work in progress. In F. L. Lin (Ed.), Common Sense in Mathematics Education: Proceedings of 2001 The Netherlands and Taiwan Conference on Mathematics Education (pp. 1–39). Taipei, Taiwan.

RME4: Webb's Opening Remarks

Opening Session - Friday, September 27, 2013

David Webb - Executive Director of Freudenthal Institute US and Associate Professor of Mathematics Education, University of Colorado Boulder

David Webb

David Webb welcomed us to the 4th International Realistic Mathematics Education Conference (#RME4) by addressing a shift in organizational structures. What used to be simply the Freudenthal Institute in the Netherlands is now the Freudenthal Institute for Science and Mathematics Education, and its American counterpart, Freudenthal Institute US, is now part of a larger CU-Boulder effort known as the Center for STEM Learning. These shifts reflect a desire to not just have cooperation between mathematics and science disciplines, but a perceived need to create innovative new STEM curricula along with the supporting frameworks, teacher education, and professional development to support it. Webb announced that earlier in the week that FISME and the Center for STEM Learning had formally agreed to collaborate, although we'll have to wait and see how this collaboration takes shape.

At its core, RME is a set of principles for curriculum design. It is sensible, then, to seek common ground in mathematics and the sciences for ideas upon which we can design curriculum. Some of that common ground is found in how we reason in math and science, and Webb offered these four activities:

• Recognition of patterns
• Making conjectures from observation
• Reasoning from evidence
• Generating new evidence

From these, we can think about how we consider the acts of modeling, problem solving, generalizing, and proving in both math and science. There are similarities and differences, and these things are meant as a starting point, not a definitive list. Perhaps the most fundamental RME principle is that of progressive formalization (see here for an example), so we must also think about how informal contexts can be used in both math and science, as well as the preformal models and representations that support more formal kinds of student thinking. Webb encouraged us to consider these RME traditions as we stretched ourselves beyond our usual disciplines, and with that the conference was underway.

Teaching Statistics: Textbook Considerations

I have the pleasure of teaching an undergraduate basic statistics class this fall for the third consecutive year. It's not a class I had any specific preparation to teach, but I've tried to make up for that by becoming familiar with some of the statistics education literature, bolstering my content knowledge (although I doubt it will ever be as wide or deep as I'd like), getting access to good resources, and being mindful of the needs of my students.

First, it would help to know a little bit about the course. Most strikingly, the class only meets once a week on a Thursday from 4:30 to 7. If you're used to teaching 180-day school years, you really have to wrap your head quickly around the idea that you're only going to see these students 15 times before finals. Also, despite the class being taught in the School of Education, it's not required of any education students. Instead, the class consists mostly of students from two majors: Sociology and Speech, Language, and Hearing Sciences. Honestly, most of them admit to avoiding math classes, but they usually need the stats class to apply for graduate school. As for the content of the course, here is how it is described in the university catalog:

Introduces descriptive statistics including graphic presentation of data, measures of central tendency and variability, correlation and prediction, and basic inferential statistics, including the t-test.

And that's it. As someone who works almost daily with the Common Core State Standards, building a course around such a sparse description would be quite a challenge, especially for a first-time instructor. When I talked to Derek Briggs about teaching the course, he advised that I use his preferred text, Statistics by Freedman, Pisani, and Purves. I'd recently used Agresti and Finlay's Statistical Methods for the Social Sciences for my qualitative methods courses, and while that book suited me pretty well, I was open to something different so I ordered the Freedman text for my class.

In hindsight, the Freedman text was fine, and the Agresti text would have been fine, too. Both were decently well-written and had plenty of problems to assign, but that's the thing — I was looking for a text that offered considerably more than explanations followed by problem sets. I really wanted something that supported students working together in groups during class, making sense of the material as we went along.

One book that had gotten my attention was Workshop Statistics: Discovery with Data by Rossman and Chance. I recognized Beth Chance's name immediately from some of the stats education literature I'd read, and felt good that this text would offer what I was looking for. I used the text last year and was not disappointed, and will be using it again this year. Below is a summary of some of the reasons I like Workshop Statistics.

Context Continuity

In the front matter of the book, Workshop Statistics contains a list of activities by application — in other words, they've categorized all the problems by context and indexed exactly where those contexts get used. The list of related problems appears again with each problem in the text (inset in picture above), so it's easy for me or my students to refer back or forward to where that context appears. I believe in teaching mathematics rooted in context when possible, so I found this an especially helpful way of finding problems that might be relevant or interesting to the students in my class.

Preliminaries

Every topic (lesson) in the text opens with some preliminary questions. Some involve data collection, which is great, but at the very least it gives students an opportunity to consider a question and how we might answer it. If Dan Meyer has made anything clear, it's that we shouldn't teach math as finding answers to questions that nobody has bothered to ask.

In Brief

The end-of-topic summary certainly isn't unique to this text, but the "You should be able to" statements are very handy for writing objectives for standards-based grading. (I hope to write about my SBG approach in a future post.)

Online Supports and Simulations

Besides both online instructor and student resources, the text uses a number of custom applets that often really help illustrate some of the concepts in the course. Some are Java, but a number have been converted to JavaScript for use on more platforms. I've avoided having students use software beyond a spreadsheet, and some of these applets have saved us from having to purchase SPSS (expensive!) or trying to use R (steep learning curve!).

Activities

The in-class activities use some interesting contexts and support groups working together. If anything they can be a bit over-scaffolded, but that relieves me from having to lecture much and I can spend most of my time going group-to-group in the classroom and dealing with questions more intimately.

Overall

There are a number of smaller things that I'm fine with, although they aren't deal-makers or deal-breakers. The pacing of the text is good — if we cover about two topics a week, we finish the text and pretty much everything one would expect in a basic statistics course. The order of the topics is sensible, too. Typically, it makes sense to put descriptive statistics before inferential statistics, and to work from one-variable stats to two-variable stats. This book is no different. Some texts put linear regression earlier, and where probability should land in a book seems to be negotiable. The placement of those topics in this book is fine for this course and the progression from topic to topic was very manageable.

Other than my first day activity, I haven't written much about teaching stats, but look for me to change that this semester.

A Genealogy of Realistic Mathematics Education

Time sinks are curious things. Some are tedious, some are frustrating, and some turn out to be fun. A few months ago, when I probably should have been studying for my comprehensive exams, a simple conversation in the office with +Ryan Grover started a genealogical journey (academically speaking) to trace back our origins and those of Realistic Mathematics Education (RME). I'm pretty good with my mathematics education history, and I knew RME in the United States took hold with the Mathematics in Context curriculum project, bringing Thomas Romberg and others at the University of Wisconsin together with Jan de Lange and others at the Freudenthal Institute at the University of Utrecht in the Netherlands. I suggested to Ryan that we search the Mathematics Genealogy Project to see if there were other ties between U.S. math ed and the history of RME, and the time sucking began in earnest.

With Ryan searching at his desk and me at the chalkboard, after several (or 4? 5?) hours we had traced back our academic roots many generations. Here's a glimpse of that work after some un-criss-crossing of arrows by Ryan, but probably still with a few mistakes:

Ryan and I left the office after dark. When I got home, my thoughts were still consumed about organizing and preserving this history, so I stayed up most of the night creating this in Google Drawings (part of Drive/Docs), where it would be easier to edit and be more shareable:

Years indicate when doctorate(s) earned. Download large version.

I've highlighted a few individuals who I think stand out. At the top left is Nicolaus Copernicus. We could have traced back a few more generations, but beginning with the person who is famous for not putting the Earth at the center of the universe seemed like a good place to start. The next one down, in yellow, is Jakob Thomasius. Although more of a philosopher, he was advised by Friedrich Leibniz and advised his more famous son, Gottfried Leibniz, and represents an early connection between the left and right sides of the diagram. The next person down, again in yellow, is Abraham Kästner. Ryan and I had never heard of him, but he's an extraordinarily connected fellow in this chart. Five of Kästner's 10 documented students are represented here, and an unseen one, to Johann Bartels, leads directly to Nikolai Lobachevsky. Furthermore, Kästner's bio on Wikipedia reads like some stereotypically tragic mathematician's drama, having been engaged to a woman for 12 years, only to marry her and see her die within the year. So then he had a daughter with his maid and spent his later years writing poetry.

If you look around, you'll find Kant, Euler, Gauss, and others, but prominently representing an early attention to mathematics education is Felix Klein, again in yellow. Klein became interested in the teaching of mathematics around 1900, and the International Commission on Mathematical Instruction (ICMI) has named their lifetime achievement award after Klein. From Klein we establish three major lines: the U.S. line through William Edward Story, a separate U.S. line to Maxime Bôcher, and a German line through Hilbert and Bieberbach. Curiously, the tree artwork on the main page of the Mathematics Genealogy Project shows the link from Klein to Story, although neither Klein's or Story's page establish a recognized advisor-advisee relationship. After checking a few other sources, it seems that making the Klein-Story connection is typical.

Now we have three major figures in the third row from the bottom. All were trained as mathematicians but transformed themselves into prominent figures and researchers in mathematics education. On the left is Ed Begle, colored in red to reflect his association with Stanford. Begle was the director of the School Mathematics Study Group, creators of what most call the "New Math" of the 1960s and 1970s. I don't want to overgeneralize, but Begle and his descendents tend to focus on the curriculum, instruction, and policy aspects of mathematics education.

Next is Henry Van Engen, colored in purple to signify his association with Iowa State Teachers College, now known as the University of Northern Iowa (my alma mater). Van Engen's publications going back to the 1940s reveal that he was focused on learning and meaning in mathematics. Instead of relying wholly on his mathematics background, he incorporated ideas from figures like Brownell and Piaget. Van Engen left ISTC in the late 1950s to help establish the math ed program at the University of Wisconsin - Madison, and his lineage of Leslie Steffe and Paul Cobb represent one of the strongest learning science traditions in math education.

On the right is Hans Freudenthal, shown in orange to signify his place in the Netherlands. A giant figure internationally, ICMI's other major international mathematics education award is named for Freudenthal in recognition of a major cumulative program of research. Whereas I associate Begle with curriculum, and Van Engen with learning science, to me Freudenthal represents a math education visionary and philosopher, someone able to reflect broadly on the field and history of mathematics and structure a new approach to mathematics education. Interestingly, Freudenthal's involvement in mathematics education was in part inspired by Begle and the New Math -- not liking what he saw in the New Math and fearing Europe would adopt a similar approach, Freudenthal steered the Netherlands in the direction we now call RME.

There are a few hidden connections at the bottom of the diagram that reflect my experience studying RME. Being at the Freudenthal Institute US and working with David Webb is the most prominent, but I greatly anticipate opportunities to learn from our FI colleagues from the Netherlands. Paul Cobb's collaboration with Koeno Gravemeijer in the early 2000s was mutually beneficial and has influenced me greatly, as Cobb's theories of learning mathematics work well in the context of RME.

Copernicus, more than 20 generations away, tends to be less influential.

---

Notes: Along the way I explored a number of other connections less related to RME and my perspective of it. William Brownell, for example, can be traced back through a number of psychologists to Kastner. Alan Shoenfeld, another mathematician-turned-math educator, can also be traced back to Kastner and has two different lines back to Gauss. Kastner seemed to turn up everywhere, while Issac Newton turned up nowhere.

Come Study With Me

Have you ever thought about pursuing a PhD in mathematics education? It's application time here at CU-Boulder and I'd like to encourage you to apply. For me, I had no idea what to expect of a PhD program, or why I should consider one program over another. I feel fortunate to have landed a spot in a high-quality program in my now-home state, but for many of our students it's worth a move across the country to be in Boulder to work with some of the best faculty you'd find anywhere. If you apply to CU's School of Education, here's some of what you should expect.

If you get accepted you should hear from us in February and be invited to a recruiting weekend in March. I'm guessing we may want to accept 2-4 math ed students this year, but that all depends on the quality of the applicants for both mathematics education and for all the other programs. At the recruiting weekend you'll meet the faculty, tour the campus, and spend time talking to your future advisor. (More on those people later.)

If you choose to attend CU-Boulder and enter the program, you'll be required to take on graduate studies full-time. You'll take most of your first-year courses as part of a cohort, and you'll have an assistantship that will likely include doing research, teaching courses, or supervising student teachers. There are many PhD programs that allow for part-time study over a decade or more to complete a degree. We're not one of them. Instead, you get to make graduate school your full-time focus, and in return the university promises to support you with an assistantship and tuition credits for at least three years. No, you won't get rich in grad school, but for many students it allows them to pay the bills and avoid racking up debt.

There are five program areas in the School of Education: Ed Psych/Learning Sciences, Foundations and Policy, Equity and Cultural Diversity, Research and Evaluation Methodology, and Curriculum & Instruction. Mathematics education fits into C&I along with science education and literacy studies. During your first year you'll be exposed to people and ideas from all the program areas as you learn the research methods and foundations that will prepare you for your future studies.

We have three faculty members in math education. Over the course of your studies you'll be advised by one or more of them, you may have an assistantship with them, and will probably take a course from all of them. Here's a little bit about each:

David Webb (my advisor) has interests in mathematics curriculum, assessment, computer science, and Realistic Mathematics Education (RME). RME is a curriculum design philosophy originating with Dutch mathematician Hans Freudenthal, and when David came to CU from Wisconsin he brought the Freudenthal Institute US with him as its Executive Director. If you study math ed at CU-Boulder, you're also studying at the Freudenthal Institute, and RME gives us a common foundation for how we conceptualize the teaching of mathematics and the design of curriculum.

Vicki Hand's primary interests are in learning theory and equity. She combines the two in powerful ways to describe how math can be taught and learned to all students. It's tough but critical work. I was fortunate to take Vicki's "Theories of Math and Science Learning" course last spring and it was one of the best courses I've had at CU. Even better, Vicki is an absolute joy to be around.

The newest addition to our math ed faculty is Edd Taylor, who will join us this spring. Not only will Edd give us the extra elementary math ed experience and knowledge we've been looking for, but his interests seem to be a natural bridge between David and Vicki's areas of expertise. Some of Edd's recent work involves working with teachers to understand and modify their curriculum to better meet the needs of culturally diverse learners. We're all excited to have him join us and I can't wait to take a class with him next semester.

While those are the main three math ed faculty, you'll encounter other faculty with knowledge or interests in math ed. For example, Kim Bunning is one of our master teachers in our CU Teach program and is herself a product of our math ed PhD program. Bill Penuel is a learning scientist with a science background, but has several projects (including one I'm on) that reach into mathematics education. Margaret Eisenhart has for a long time studied how to interest females and minorities in STEM fields. If you have any interest in statistics and quantitative research methods, we have some of the best faculty anywhere, including Derek Briggs, Greg Camilli, Andrew Maul, and our Dean, Lorrie Shepard. So while we might not have the biggest program around, we make up for it with high quality experiences and high quality people school-wide.

Last but not least are the other math ed graduate students. Of those who aren't busy finishing their dissertations, you'll come to know:

• Michael Matassa, a former middle school math teacher and instructional coach who teaches elementary methods and is interested in researching mathematics teaching;
• Bill Campbell, a former elementary teacher with broad educational interests and knowledge whose research interests lie in RME and elementary math education;
• Louisa Harris, a math-turned-math ed doctoral student interested in the challenges for women studying math in college and graduate school;
• Ryan Grover, another math-turned-math ed doctoral student interested in RME approaches in undergraduate mathematics;
• Ian Her Many Horses, a former computer science and math teacher who's pioneering a route in the emerging field of computer science education;
• Fred Peck, a former high school math teacher who's interested in better understanding how students progress from informal to formal mathematical understanding; and
• Vinnie Basil, a former science and math teacher who is interested in educational equity and integrated approaches to math and science curriculum.

So How Do I Apply?

The application process isn't horribly difficult, but you'll have to act quickly if you don't already have application materials put together. You'll need things like your GRE scores, reference letters, transcripts, and a personal statement. Although the personal statement isn't very long, it's your opportunity to show that you have the kind of writing skill that you can later apply to your dissertation. That's important! And of course, as a math ed student, we expect your GRE math score to be pretty good. Applications are due January 1, and you should follow these instructions and not be afraid to ask questions!

OpenComps: Candidate Status Unlocked. Loading Next Level...

I'm more than a week tardy in reporting this, but my oral examination went well and I've transitioned from "PhD student" to "PhD candidate." In other words, I passed my comprehensive exams. Apparently the title isn't universal (I've heard some schools progress you from "candidate" to "Candidate," changing only the capitalization), but what it means is that my focus and responsibility shifts away from coursework and onto my own research.

Normally this means I'd be taking few, if any, classes next semester and working on a prospectus. But as luck would have it, the School of Education is chock-full of great course offerings in the spring. So I'll be taking a full slate of courses: Language Issues in Education Research, Research on Teaching and Teacher Education, and Advanced Topics in Mathematics Education. Throw in our departmental seminar and the five dissertation hours we're required to carry each semester, and it looks like I'll be scheduled for 15 credit hours. Which is a lot for a doctoral student, er, candidate.

Realistically, this means my prospectus will probably wait until summer. That shouldn't be an inconvenience. It's going to take me a while to focus in on a research question anyway, and I think a combination of working on Bill Penuel's Inquiry Hub project and taking the Research on Teaching and Teacher Education class with Dan Liston and Jennie Whitcomb will give me plenty to think about. I am very interested in issues of research to practice, which means I need to look more at Paul Cobb's latest work, Cynthia Coburn's work, and keep working with Bill on Design-Based Implementation Research. I also want to learn more about how and why teachers modify their curriculum, which means getting up-to-date with the work of people like Janine Remillard and Corey Drake. The better I understand the current boundaries of work in these areas, the better I'll know what direction my work should go.

OpenComps: Written Exam Down, Oral Exam to Go

About two weeks ago I submitted my written responses to my comprehensive exam questions. I can't go into detail about the questions, but I'll summarize them this way:

1. Here's a dichotomy from the learning sciences. Deal with it.
2. Somebody did a quantitative study X and now wants to do Y. Before you think Y is a good idea, what do you have to know about X?
3. How would you help math teachers learn about X given conditions Y?

I hadn't quite anticipated Question 1 so there was some background work to do before I could address certain details. Thankfully, I was pretty well prepared to structure my argument, and it was on this question that I did my best writing. While I'd had dreams of finishing a couple questions before the end of the weekend, my actual pace was slower than that. A lot slower. By the end of Friday, I'd written about a paragraph, and by the end of Saturday, I'd written about a page. Fortunately, that was the foothold I needed to have the rest of the 9-page paper finished on Sunday.

Next I answered Question 2. In some ways this was the question that worried me the most, but my studying definitely helped. Still, my writing was slow and it wasn't really until late Wednesday when I had this question finished. When you have three questions to answer in seven days, taking six days to answer the first two questions is less than ideal.

That left me to answer Question 3 in a bit of a writing sprint starting in the wee hours of Thursday morning, breaking to attend and teach class Thursday afternoon and evening, and then writing until 7am Friday morning to finish. Question 3 was my advisor's question and the one for which I was most prepared; in fact, a couple pages was largely a rehash of some of some things I'd blogged about in the past. Having that for a strong start certainly helped the rest of the paper take shape rather quickly.

It was a relief to reach the end of comps week, but I couldn't get too much rest because I had put off a number of things (okay, almost everything) during comps and in the weeks leading up to comps. Professors and fellow students are very understanding about it, which is great, but I wasn't entirely comfortable using comps as an excuse to not do much else during that time. In the past two weeks (including some of every day of my fall break), I've been catching up with the class I take, the class I teach, and the research project I'm on. I haven't been blogging and my social media activity has been pretty minimal during this time, but I'm starting to feel caught up.

The last hurdle to clear is the oral examination, scheduled for this Tuesday morning. I'm not too concerned about it, and thankfully, the message from my comps committee has been not to worry. But between now and then I will be going back over my responses, double-checking the literature I cited, and reading a few new things I uncovered during the comps process. My advisor hinted at some things he wants to talk about and I'll be sure to prepare for those things, too.

OpenComps: Final Preparations

By 9 am Friday, November 2nd, my advisor will email me my three comprehensive exam questions. I have exactly a week to answer them. He says I'm prepared, and I appreciate his confidence in me. I think I'm reasonably prepared, too, and I greatly appreciate that among my numerous anxieties, test-taking isn't one of them. Far from it, in fact. See, I'm one of those mystical kids that policymakers have in mind when they come up with laws like No Child Left Behind. I'm the one who actually likes taking tests and fools himself into thinking they're just a harmless yet useful snapshot of broad academic knowledge and skill. Give me a #2 pencil and bubbles to fill in and I'll happily work for hours.

There won't be any bubbles on my comprehensive exam, but there will be hours of work. Over the past week I've been making my final preparations, most of which are designed to make next week go as smoothly as possible. A summary:

Ready My References

I think my personal library will have most of the math and learning science books I might want, but I felt like some extra perspectives and guides concerning experimental design, casual inference, and statistics might come in handy. I know I can't expect to read any of these cover-to-cover in the course of the next week, but if nothing else the examples and explanations they contain could be valuable.

Having books around is a luxury, but for this level of work, it's even more important to have a way of keeping track of the hundreds of journal articles that I might want to use in my comps responses. I've been using Mendeley as my reference manager since the spring of 2010. Regardless of what tool you use -- Zotero, RefWorks, Endnote, Papers, etc. -- it's important during any writing period to have something that allows you to focus on writing, not scrambling for citation information and digging through the APA style book.

One of the best investments I've made as a grad student has been my diligent attention to the annotation, curation, and metadata accuracy of my Mendeley library. I was somewhat lax about it during my master's year and my first year of the PhD program, but then I spent most of two weeks of a summer going back through every PDF, every book, every syllabus, and every paper I wrote to make sure I had everything neatly cataloged. And I haven't relaxed since. Right now I have 890 references in my personal library, with others in group collections, and I can find or cite any of them in just seconds.

If there's one thing I can't let myself do is turn my comps into a massive search for new literature. I admit, I love the thrill of the hunt, and I've spent many hours digging around in Google Scholar tracking down papers that I realistically have no time to read. I need to trust that most of what I need I already have and I've already read, and keep my literature hunting to a minimum.

Minimize Distractions and Get Comfortable

For my last week before comps, I actually spent very little time studying and more time minimizing potential distractions. I've been to the grocery store, I've washed dishes and laundry, and I reformatted and reinstalled my operating system, virtual machine, and software on my computer because a few things had gotten flaky after a year of hard use. I've never liked studying right before a test anyway, as any attempt to "cram" is nullified by thoughts that always begin, "If I don't know it by now...." I passed my 100-hour studying mark a week or so ago and that will have to be good enough.

I'll probably work mostly at my desktop. If your computer had three monitors, 16 GB of RAM, university broadband peaking at nearly 90Mbps up and down, and a pair of Sennheiser HD 595s, you'd probably work at it, too. I might try working some in my office, and my kitchen table is nice for when a lot of open books are involved. I don't want to be stuck in my office chair for 18 hours a day, so I plan to do some heavy thinking while running and in a pinch I can even prop my laptop up on my exercise bike.

Sometimes I work in silence, but not very often. I don't want to get distracted by moving pictures, but there are a few movies I can play for background noise without losing focus, mostly because I've seen them so many times. I can get distracted by podcasts, so I'll try to listen to those selectively over the next week. I'll listen to a lot of music, and my tastes for a task like this tend to be towards the incredibly gifted (Tori Amos, Curtis Mayfield, Norah Jones, Sia) and music that's downtempo/trip-hop or otherwise having a likable female vocal/bass combo (Thievery Corporation, Zero 7, Garbage). Seriously, in the midst of an important exam, who wouldn't want to perform as relaxed and confidently as LouLou?:

OpenComps Get Less Open

Obviously, yet unfortunately, once I get my questions I'm pretty limited in what I can say about them. I'm not to receive outside help, solicited or unsolicited, and even after the exam is over I'm only to talk about the process in general terms. (I'm assuming that's in the event my committee members want to reuse the same or similar questions in the future.) Assuming I'm not exhausted by the process, I'll try to summarize my approach and workflow, lessons learned, and hopefully some epiphanies that come in the process of working through my questions. I'm looking forward to the week of writing and then readying myself for the oral defense, scheduled for November 27th.

OpenComps CGI

No, I don't mean "computer-generated imagery." Or the "Clinton Global Initiative." Or "Common Gateway Interface." In the world of mathematics education, CGI stands for "Cognitively Guided Instruction," one of the most robust lines of research produced in the past several decades. If you study math education, you're probably going to study CGI. If you study math education and your advisor is from the University of Wisconsin, then you're definitely going to study CGI. Here's my reading list:

Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97(1), 3–20. doi:10.1086/461846

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C.-P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499–531. doi:10.3102/00028312026004499

Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15(3), 179–202. doi:10.2307/748348

Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27(4), 403–434. doi:10.2307/749875

Franke, M. L., Carpenter, T. P., Levi, L., & Fennema, E. (2001). Capturing teachers’ generative change: A follow-up study of professional development in mathematics. American Educational Research Journal, 38(3), 653–689. doi:10.3102/00028312038003653

Knapp, N. F., & Peterson, P. L. (1995). Teachers’ interpretations of “CGI” after four years: Meanings and practices. Journal for Research in Mathematics Education, 26(1), 40–65. doi:10.2307/749227

This works out nicely because CGI also happens to be a topic of discussion this week in my "Advances in Assessment" class. (Related note: Due to Erin Furtak being out of town, Lorrie Shepard will be our "substitute teacher." That leads to the natural question: Great sub, or greatest sub?) CGI was also featured prominently in Randy Philipp's NCTM Research Handbook chapter on teacher beliefs and affect. Even though my knowledge of CGI is limited, I sense that lines of research like CGI are the stuff math education researchers dream about: long-lasting, productive, well-funded areas of study that help both students and teachers in measurable and meaningful ways.

OpenComps Study of Teacher Beliefs; MathEd.net Turns Three

A month from now I'll be in the midst of the written portion of my comprehensive exam. My last #OpenComps update (and several posts since then) listed several readings about teacher learning. With those complete, now I'm moving my attention towards teacher beliefs with the following articles and chapters:

Fennema, E., & Franke, M. L. (1992). Teachers knowledge and its impact. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 147–164). Reston, VA: National Council of Teachers of Mathematics.

Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307–332. doi:10.3102/00346543062003307

Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257–315). Charlotte, NC: Information Age.

Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). Reston, VA: National Council of Teachers of Mathematics.

Villegas, A. M. (2007). Dispositions in Teacher Education: A Look At Social Justice. Journal of Teacher Education, 58(5), 370–380. doi:10.1177/0022487107308419

Wilkins, J. L. M., & Brand, B. R. (2004). Change in preservice teachers’ beliefs: An evaluation of a mathematics methods course. School Science and Mathematics, 104(5), 226–232. doi:10.1111/j.1949-8594.2004.tb18245.x

As I usually do, I'm reading these in chronological order. I just finished the Pajares article and will next move on to Alba Thompson's well-regarded chapter from the 1992 NCTM research handbook. My advisor said I probably don't need to read the entire Fennema & Franke chapter, but there is a diagram near the end that I should be aware of and the context surrounding it.

MathEd.net Turns Three

Although I've been blogging my random thoughts and personal commentary since 2001, after starting graduate school I knew I'd be blogging more about education. Three years ago today, I decided it was time to split my identity: one blog and Twitter account for professional/educational content, and a separate blog and Twitter account for personal/miscellaneous content. It's been a good decision, one that has spared many of you from numerous updates about the Cubs, college wrestling, or my infrequent travels.

I'm creeping up on 40,000 page views, which I think is pretty good given how infrequently I sometimes post and how technical some of what I'm writing about has become. It reminds me largely of why I started this blog: as a teacher, I was willing to have my practice improved by knowledge from research, if only I could find it. The research literature was locked behind paywalls I couldn't afford, and as a lone math teacher in a rural district, I didn't have instructional coaches or curriculum staff to help me. But I knew smart people and resources existed online, and that social tools were allowing us to come together in new ways. The best ticket for admission in that social world is one's own contributions, and I'm trying to contribute something not easily found elsewhere.

I thank you all for reading, and I look forward to what the future brings -- not only for this blog and for myself, but also where this disintermediated online sharing of educational knowledge might take us.

OpenComps Update

With five weeks to go before beginning the written portion of my comprehensive exam, I recently met with my advisor to discuss gaps in my reading list. I think everybody has holes somewhere in their knowledge, but given my interests in research and practice we came up with additional readings focused on three areas: teacher learning, teacher beliefs, and cognitively guided instruction (CGI). I'm starting with teacher learning, which includes the following four articles:

Ball, D. L. (1988). Unlearning to teach mathematics. For the Learning of Mathematics, 8(1), 40–48. Retrieved from http://www.jstor.org/stable/40248141

Ball, D. L. (2008). Content knowledge for teaching: What makes it special ? Journal of Teacher Education, 59(5), 389–407. doi:10.1177/0022487108324554

Lampert, M. (2009). Learning teaching in, from, and for practice: What do we mean? Journal of Teacher Education, 61(1-2), 21–34. doi:10.1177/0022487109347321

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Retrieved from http://www.jstor.org/stable/3202180

Although I have a vague understanding of pedagogical content knowledge (PCK) and mathematical knowledge for teaching (MKT), I knew I needed to dig into Shulman's and Ball's thoughts to better understand their origins. In a way, it's a pretty good sign when the gaps you perceive yourself as having are more or less the ones your adivsor sees, too. There are places for life to contain wonderful surprises, but I don't think this needs to be one of them. Now, on to the reading!

RYSK: Shepard's The Role of Assessment in a Learning Culture (2000)

This is the 13th in a series describing "Research You Should Know" (RYSK).

In her presidential address at the 2000 AERA conference, Lorrie Shepard revealed a vision for the future of educational assessment. That message turned into an article titled The Role of Assessment in a Learning Culture, and its message is still very much worth hearing today. Lorrie Shepard remains a globally-respected expert in assessment, psychometrics, and their misuses, and I'd think she was totally awesome even if she wasn't my boss.

Shepard is often present for debates about large-scale testing, but this paper focuses on classroom assessment -- the kind, says Shepard, "that can be used as a part of instruction to support and enhance learning" (p. 4). Shepard does this by first explaining a historical perspective, then describing a modern view of learning theories, then envisioning how new assessment practices could support those theories. Impressively, she does this all in just 11 well-written pages. (In fact, given that the paper is available on the web, I wouldn't blame you at all for skipping this summary and just reading the article for yourself.)

History

Shepard highlights several major themes from history that have continued to drive our assessment practices. One is the social efficiency movement, which "grew out of the belief that science could be used to solve the problems of industrialization and urbanization" (p. 4). While this movement might have helped our economic and educational systems scale rapidly (think about Ford and the assembly line), social efficiency carries with it a belief that people have a certain innate (and largely fixed) set of capabilities, and our society operates its most efficiently when we measure people and match their capabilities to appropriate education and employment. For example, students were often given IQ tests to determine if their future path should lie on a particular academic or vocational track.

The dominant learning theories of the early and mid-1900s were associationism and behaviorism, both of which promoted the idea that learning was an accumulation of knowledge that could be broken into very small pieces. Behaviorism was also tied closely to theories of motivation, as it was believed learning was promoted when knowledge was made smaller and opportunities for positive reinforcement for learning were made greater. Much of the assessment work related to these beliefs can be traced back to Edward Thorndike, considered to be the father of scientific measurement and earliest promoter of "objective" testing. It's been 100 years since Thorndike was elected president of the American Psychological Association, and decades since his ideas seriously influenced the leading edges of learning theory. Still, as most anyone who works in schools or experienced a traditional education can attest, ideas of social efficiency and behaviorism are still evident in schools -- especially in our assessment practices.

Together, the theories of social efficiency, scientific measurement, and beliefs about intelligence and learning form what Shepard sees as the dominant 20th-century paradigm. (See page 6 of the paper for a diagram.) It's important to begin our discussion here, says Shepard, because "any attempt to change the form and purpose of classroom assessment to make it more fundamentally a part of the learning process must acknowledge the power of these enduring and hidden beliefs" (p. 6).

Modern Theories

In the next section, Shepard describes a "social-constructivist" framework that guides modern thought on learning:

The cognitive revolution reintroduced the concept of mind. In contrast to past, mechanistic theories of knowledge acquisition, we now understand that learning is an active process of mental construction and sense making. From cognitive theory we have also learned that existing knowledge structures and beliefs work to enable or impede new learning, that intelligent thought involves self-monitoring and awareness about when and how to use skills, and that "expertise" develops in a field of study as a principled and coherent way of thinking and representing problems, not just as an accumulation of information. (pp. 6-7)

These ideas about cognition are complimented by Vygotskian realizations that the knowledge we construct "is socially and culturally determined" (p. 7). Unlike Piaget's view that development preceded learning, this modern view sees how development and learning interact as social processes. While academic debates remain about the details of cognitive vs. social (and vs. situative vs. sociocultural vs. social constructivist vs. ...), for practical purposes these theories can coexist and are already helping teachers view student learning in ways that improve upon behaviorism. However, Shepard says, since about the 1980s this has left us in an awkward state of using new theories to inform classroom instruction, while still depending on old theories to guide our assessments.

Improving Assessment

If we wish to make our theories of assessment compatible with our theories of learning, Shepard says we need to (a) change the form and content of assessments and (b) change the way we use and regard assessment in classrooms. Some of the potential changes in form are already familiar to most teachers, such as a greater use of open-ended performance tasks and setting assessment tasks in real-world contexts. Furthermore, Shepard suggests that classroom routines and related assessments should reflect the need to socialize students "into the discourse and practices of academic disciplines" (p. 8) as well as foster metacognition and important dispositions. Shepard does not go into much more detail here because others have already given attention to these ideas, but gives us this simple yet powerful idea (p. 8):

"Good assessment tasks are interchangeable
with good instructional tasks."

Next Shepard pays special attention to negative effects of high-stakes testing. Shepard could be called a believer in standards-based education, but recognizes how "the standards movement has been corrupted, in many instances, into a heavy-handed system of rewards and punishments without the capacity building and professional development originally proposed as part of the vision (McLaughlin & Shepard, 1995)" (p. 9). Unfortunately, Shepard's predictions have held true over the past 12 years: we've seen test scores distorted under political pressure, a corruption of "teaching to the test," and a trend towards the "de-skilling and de-professionalization of teachers" (p. 9). What's worse might be a decade of new teachers who've learned to "hate standardized testing and at the same time reproduce it faithfully in their own pre-post testing routines" (p. 10) because they've had such little exposure to better forms of assessment.

For the rest of the article, Shepard focuses on how assessment can and should be used to support student learning. First, classrooms need to support a learning culture where "students and teachers would have a shared expectation that finding out what makes sense and what doesn't is a joint and worthwhile project" (p. 10). This means assessment that is more informative and reflective of student learning, one where "students and teachers look to assessment as a source of insight and help instead of an occasion for meting out rewards and punishments" (p. 10). To do this, Shepard describes a set of specific strategies teachers should use in combination in their classrooms.

Dynamic Assessment

When Shepard wrote this article, formal ideas and theories about formative assessment were still emerging and the field had yet to settle on some of the language we now use. But if you're at all familiar with formative assessment, Shepard's description of "dynamic" assessment will sound familiar: teacher-student interactions continuing through the learning process rather than delayed until the end, with the goal of gaining insight about what students understand and can do both on their own and with assistance from classmates or the teacher.

Prior Knowledge

The idea of a pre-test to see what students know before instruction begins is not new, but Shepard says we should recognize that traditional pretests don't usually take account of social and cultural contexts. Because students are unfamiliar with a teacher's conceptualization of the content prior to instruction (and vice versa), scores might not accurately reflect students' knowledge as well as, say, a conversation or activity designed to elicit the understandings students bring to the classroom. Also, as Shepard has frequently observed, traditional pre-testing often doesn't significantly affect teachers' instruction. So why do it? Instead, why not focus on building a learning culture of assessment: "What safer time to admit what you don't know than at the start of an instructional activity?" (p. 11)

Feedback

The contrast in feedback under old, behaviorist theories and newer, social-constructivist theories is clear. Feedback under old theories generally consisted of labeling answers right or wrong. Feedback under new theories takes greater skill: teachers need to know how to ignore student errors that aren't immediately relevant to the learning at hand, while crafting questions and comments that force the student to question themselves and any false knowledge they might be constructing. (See Lepper, Drake, and O'Donnell-Johnson, 1997, for more on this.)

Transfer

While it is our hope that our students will be able to generalize the specific knowledge they have learned and apply it to other situations, our ability to accurately research and make claims about knowledge transfer turns out to be a pretty tricky business. Under a strict behaviorist perspective, it was appropriate to believe that each application of knowledge should be taught separately. Many of our current theories support an idea of transfer, and evidence shows that we can help students by giving them opportunities to see how their knowledge reliably works in multiple applications and contexts. So while some students might not agree, Shepard says teachers should not "agree to a contract with our students which says that the only fair test is one with familiar and well-rehearsed problems" (p. 11).

Explicit Criteria

If students are to perform well, they need to have clear guidance about what good performances look like. "In fact, the features of excellent performance should be so transparent that students can learn to evaluate their own work in the same way their teachers would" (p. 11). This reinforces ideas of metacognition and, perhaps more importantly, fairness.

Self-Assessment

There are cognitive reasons to have students self-assess, but other goals are to increase student self-responsibility and make teacher-student relationships more collaborative. Students who self-evaluate become more interested in feedback from others, are more aware of standards of excellence, and take more ownership over the learning process.

Evaluation of Teaching

This is another idea now heavily intertwined with formative assessment, but Shepard takes it one step farther than I normally see it. Instead of just using assessment to improve one's teaching, Shepard recommends that teachers be transparent about this process and "make their investigations of teaching visible to students, for example, by discussing with them decisions to redirect instruction, stop for a mini-lesson, and so-forth" (p. 12). This, Shepard says, is critical to cultural change in the classroom:

If we want to develop a community of learners -- where students naturally seek feedback and critique of their own work -- then it is reasonable that teachers would model this same commitment to using data systematically as it applies to their own role in the teaching and learning process. (p. 12)

Conclusion

Shepard admits that describing this new assessment paradigm is far easier than it is to implement in practice. It relies on a great deal of teacher ability and confronting some long-held beliefs. Shepard recommended a program of research accompanied by a public education campaign to help citizens and policymakers understand the different goals of large-scale and classroom assessments. Neither the research or educating the public is easy, because both are built upon a history of theories and practice that a new paradigm needs to discard. Perhaps we haven't taken on this challenge with the effort and seriousness we've needed, and I worry that now we're more apt to talk about "learning in an assessment culture" rather than the other way around, as Shepard titled this article. I sometimes wonder if she's considered writing a follow-up with that title, or if she's hoping she'll never have to. I guess the next time it comes up I'll have to ask her.

Math note: This is an article about assessment and not specific to mathematics, but I'd be remiss if I didn't share Shepard's inclusion of one of my all-time favorite fraction problems:

References

Lepper, M. R., Drake, M. F., O'Donnell-Johnson, T (1997). Scaffolding techniques of expert human tutors. In K. Hogan & M. Presley (eds.), Scaffolding student learning: Instructional approaches & issues. Cambridge, MA: Brookline Books.

McLaughlin, M. W., & Shepard, L.A. (1995). Improving education through standards-based reform: A report of the National Academy of Education panel on standards-based educational reform. Stanford, CA: National Academy of Education.

Shepard, L. A. (2000). The role of assessment in a learning culture. Educational Researcher, 29(7), 4–14. doi:10.2307/1176145

Thompson, P. W. (1995). Notation, convention, and quantity in elementary mathematics. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 199-221). New York: State University of New York Press.

Scholarly Reading Strategies

While I welcome greater diversity in higher education, I recognize graduate studies aren't for everybody. More specifically, I'd suggest you think twice about a PhD if you're the kind of person who doesn't like to read. The written word is the stuff on which academia survives and thrives, and as such many more scholarly words are produced than any one person could possibly read. But yet our work depends on reading huge chunks of scholarly literature.

I was only a few weeks into my first semester as a PhD student when I realized that there were going to be times when I couldn't finish all of the assigned readings for class. Thankfully, the ever-kind Elizabeth Dutro addressed this problem in class and told us all that this was okay. Yes, sometimes there were things we'd need to understand in great detail, but other times it was enough to just gain familiarity with an article in case we needed to refer to it later. Some readings (for me, Foucault comes to mind) need multiple readings before they make any coherent sense.

I discussed this with my advisor at the time, Finbarr (Barry) Sloane. Knowing that he was a voracious reader with incredible retention and memory (Vicki Hand once told me she wished her internet connected directly to Barry's brain), I asked if he had any special reading strategies. This is essentially what he told me:

I read things three times. The first time I just read and get a sense for the article. The second time I read for details, take notes, and make connections. On the third reading, I read the article out-of-order. If I can read paragraphs or sections at random and understand them without having to re-read the surrounding context, then I know I understand it.

Now I was understanding why Barry's knowledge of the literature was so strong. Unfortunately, I was also understanding why he routinely only got a few hours of sleep every night -- all that reading and re-reading takes time. He wasn't shy about his love of reading; he said that while in graduate school in the mid-1980s, he read every single article in the Journal for Research in Mathematics Education since its first publication in 1970. That's intense.

Maybe I can't read every JRME article three times between now and my comprehensive exams, but I do need to make the most of my comps readings. So long as the quantity of reading doesn't overwhelm me, my three-part strategy will be (a) read, (b) read for detail and take notes, and (c) blog a summary. That's the approach I took with my last post and I felt very good about it. (It helped that the Clements & Sarama article was less than 10 pages long.) The written part of my comprehensive exam gives me a week to answer three questions with essays/reports of about 8-10 pages each. I figure the more I've written on my blog, the more prepared I'll be to write for comps. There's also the side benefit of giving my advisor a convenient way to keep up with my preparation while he's traveling during his sabbatical this semester. I'd love to blog about at least four or five readings a week, and you'll be the first to know if I can keep up that pace.

Project OpenComps

This semester I'll be taking my comprehensive exams, or "comps." As a first-generation college student from the working-class rural Midwest, this is pretty unknown territory for me. I remember being a naive undergraduate who had to ask what masters and doctorates were, and when I started my PhD program I had to ask similarly naive questions about the mysterious and vaguely threatening-sounding comps. Quite simply, comps is my opportunity to show a committee of faculty members that I have the knowledge and skills to take on my own research -- namely, my dissertation. Yes, there are written and oral examinations, but it's the process of working with a committee of faculty to both narrow my focus and double-check that I know what I should know that makes the process valuable.

Thankfully, I've been able to watch other graduate students prepare for and take their comps (usually passing, but not always) and now it's time to prepare for mine. I'm going to share that process and preparation with you and tag things #opencomps along the way. You might consider this a step in the direction of something like Hack the Dissertation. I've learned that the entire comps process can vary from program to program, so I can only really describe what it's like for a math education student in CU-Boulder's School of Education. Let's recap how I got this far:

• With a BA in Mathematics (Teaching) from the University of Northern Iowa and six years teaching high school math, I decided to go to grad school. Having missed the admissions deadline for a master's program, I spent a fall semester as a continuing education student and was admitted into CU-Boulder's master's program for the spring. It went remarkably smoothly, thanks to the help of my advisor David Webb.
• I expressed an interest in the PhD program and was encouraged to apply. I got recommendations from my current professors and good (enough) GRE scores to be accepted. This meant abandoning the master's program, but thankfully many of the credits I earned transferred to the PhD program.
• My first year in the PhD program was spent in the "core," the set of six courses every cohort of incoming doctoral students take in the School of Ed. Those courses include two semesters of quantitative methods, two semesters of qualitative methods, a course on theoretical perspectives on social science research, and a course on education research and policy. I took a seventh core course, covering multicultural education, the first semester of my second year.
• I focused the rest of my second-year coursework on two areas: math education (a course on algebra and a course on theories of mathematical learning) and educational measurement (a course on survey research with an introduction to item response theory, and an advanced measurement course with more IRT and generalizability theory).

I'm required to have 56 hours of coursework (not including dissertation credits) for a PhD. In some programs you need to finish those classes before comps, but in my area it's okay to just be close to 56 so long as the coursework provides the necessary foundation. With over 50 credits under my belt my advisor says I'm ready, so I've taken the first two steps this semester towards comps. First, I needed to choose a committee of three faculty members. My first choice was easy -- my advisor David Webb. I can trust David to make sure I'm ready in the areas of math education and classroom assessment. Also on my committee is Derek Briggs, who will surely hold me to task in the area of quantitative methods, validity, and causal inference. Derek didn't actually teach my core quantitative classes, but I took my measurement courses from him and enjoyed working with him. Due to my wandering interests, the third choice wasn't so easy. (Someone in policy? Qualitative methods? Stats ed? Learning sciences?) I went a bit onto a limb and chose someone I've never taken a class from: Bill Penuel. I got to know Bill a bit last spring during some facilities work, and I'm working for him this semester on a project that combines many of my interests: math ed, professional development, technology, pedagogy, task design, and assessment. I like what I've seen of the project so far and think working more closely with Bill will be a very good thing.

The second step I've taken towards comps this semester was to assemble a reading list. Basically, the reading list contains what I've read for my classes and what I've cited in papers and it gives my committee a place to look for holes in my knowledge. Thanks to Mendeley and careful curation over the past two years, the list wasn't too difficult to assemble. It's long and looking at the 40+ pages of references made me not feel so bad about not reading much over the summer. Take a look at my reading list for yourself, and feel free to ask about anything there, or suggest something you think might interest me!

A First Day Statistics Activity

I have the honor of again teaching our undergraduate statistics course in the School of Education, better known here as EDUC 4716 Basic Statistical Methods. Perhaps the most interesting thing about the course is that it's not required for any education programs, minors, or certificates. Instead, the course attracts students largely from the Department of Speech, Language, and Hearing Sciences (who don't need it to graduate, but do need it to apply to grad school or, more recently, to get certified) and sociology majors. So how does this course end up in the School of Ed? Probably due to the legacy we have in quantitative methods, thanks to people like Robert Linn, Gene Glass, Lorrie Shepard, and now faculty like Derek Briggs and Greg Camilli. Somehow all of their hard work and success filters down and gives a relative stats-hack like me a chance to teach undergrads.

Many of my students are upperclassmen and have spent much of their college experience avoiding math courses. In fact, on last year's FCQ (Faculty Course Questionnaire) my students' average rating for the item "Personal interest in this subject prior to enrollment" was a 1.8 out of 6 -- a response the university tells me is at the 0th percentile across campus. I like to think of this as a great opportunity in a "nowhere to go but up" kind of way, a chance for me to change the way students think of mathematics and see themselves as mathematical beings. Then again, it's hard to make big changes in only 15 class meetings of 2.5 hours each. If I'm going to make a difference, class has to get off to a solid start.

My opening activity this year started with the preparation of four simple index cards with different distribution shapes:

Four common distributions, clockwise from top left: normal, left skewed, right skewed, and normal.

I have the benefit of a small class of 14 students. So I cut my graphs into a total of 14 pieces:

14 pieces for 14 students. Note on the bottom I've provided the hints A, B, C, and D.

When class started, I mixed up the graph pieces and handed one to each student. Then I told the class to find the other people in class who had the graph pieces that aligned with theirs. Once they had a completed graph, form a group at one of the tables and discuss which of the following they thought their group's graph might describe:

• People born each month of the year
• Student GPAs at this university
• Student heights at this university
• Starting salaries of new graduates from this university

It took my class less than 3-4 minutes to find their groups and then I gave them another 3-4 minutes to discuss what their graph shape might describe. As a class, I had each group share their ideas and then we discussed them. Not everybody agreed initially about which shape matched which description, which led into important comments about how we might think about unbiased sampling of students and imagining different scales and labels along the horizontal axes.

So in less than 15 minutes I combined group-making, statistics, and active, student-centered problem solving into one activity. This activity also gets students thinking about distribution shapes, which I sometimes worry we ignore in the rush to calculate centers and spreads. If you're wondering how to adapt this for your classroom, I offer these suggestions:

• If you have a few more students, cut more slices.
• If you have twice as many students, consider making two of each distribution shape and scaling the x-axis to match one of 8 potential descriptions. (i.e., a normal distribution scaled for heights in inches could be distinguished from one scaled for SAT scores.)
• If you want to use this for Algebra 1, you can make graphs that describe things like, "Toni walked to the bus stop at 2 mph, rode the bus at 30 mph to the bike shop, then rode a bike back home at 12 mph." Such an activity begins CPM's Algebra Connections and was the inspiration for my activity.
• If you want to use this for Algebra 2 or higher, you can use graphs of functions that students will become familiar with (parabolas, cubics, hyperbolas, etc.). I don't think it's worth fretting over vocabulary at this point -- just give students an opportunity to think about how the functions behave and what phenomena they could possibly model.

The Publication Paradox

This week is Open Access Week (follow #oaweek on Google+ and Twitter), and while I've shared a few links and talked to some of my officemates, I haven't taken (or had, really) time to expand on my thoughts more fully. But I take Open Access very seriously, and I know the status quo (researchers signing over copyright to journals who lock away the research behind paywalls) won't change unless more of us keep sharing openly to the widest audience possible. Because the antithesis of Open Access isn't copyright -- it is the unwillingness to share any ideas at all.

In the field of education, particularly in education policy, research is conducted and published in one of two ways: either by academics to submit to journals, or by think tanks and other groups who generally do non-university-based research. Academics will defend their system because their research is peer-reviewed, whereas much think tank research is not. In fact, in an effort to force a peer review process onto think tank research, the National Education Policy Center created the Think Tank Review Project, which includes reviews of think tank research and the annual Bunkum Awards. (Disclaimer: I know, work with, and take classes from various scholars at the NEPC.) If the academics are right, and their peer-reviewed research is superior, does that mean it is more influential? Hardly. According to this research by Holly Yettick (also affiliated with the NEPC), university-based education research is only cited about twice as often in major news outlets as research from think tanks, even though universities publish about 15 times more research (2009, emphasis mine). Yettick's conclusions to this report include a recommendation to education reporters, urging them to consider more sources because "Unlike think tank employees, university professors generally lack the incentives and resources to conduct public relations campaigns involving outreach to journalists" (p. 15). My question is this: How does copyright and traditional publishing affect this incentive structure, and how can open access change it?

First, imagine you work at a think tank and you're proposing research. Even before writing a word, you probably have an audience in mind that you'd like to reach with your work. Once your research is approved, you go about the research process and publish a report. Because the think tank does the researching and the publishing, no transfer of rights are necessary -- the work was a work for hire and copyright belonged to the think tank from the very beginning. Now the think tank can set about trying to promote the results to the research to the media and other interested audiences. They have an incentive to promote because the research, the publication, and the promotion is carried out by the think tank, an organized unit that includes you in its shared ownership of the work. This gives the think tank a collective interest in spreading their ideas.

Now imagine you're a researcher at a university. You too have an audience in mind that you'd like to reach, but when your research is finished you submit your report to a peer-reviewed journal. In order for the journal to publish (or sometimes, even to consider) your article, you must transfer to them your copyrights. The journal now owns the report, and this is where the incentive system starts to break down. The article might be read by your peers, and may help you receive tenure, but surely (I hope) your peers and tenure committee don't comprise the true scope of your target audience. If you, the researcher, are still intent on making sure your work reaches the intended audience, how effectively can you promote something you no longer own? Most efforts to share your report will violate the publisher's copyright. You could create derivative work, in the form of conference presentations, blog postings, or articles for magazines, but this actually requires extra effort to avoid a copyright violation, impedes future progress on other research, and often does not count towards tenure.

Instead of self-promotion, can you, a researcher, count on a journal to promote your work? Why would they? Do they know the scope of the audience you would like to reach? What incentives does the journal have to promote work they did not create? The journal wants subscribers, to be sure, but because they have no rights to your future research (or that of any scholar), their main incentive is to preserve a system that positions their journal as one of the few credible outlets for research. For example, the American Education Research Association has 25,000 members and publishes six peer-reviewed journals. If you're an education researcher, you probably belong to AERA and you respect and read the scholarship in their journals. But in Holly Yettick's dissertation research, searching through "nearly forty thousand articles in hundreds of publications" (2011, par. 14), she has yet to see a single AERA-published article mentioned anywhere. So while you might hear Brian Williams start a story on the NBC Nightly News with the phrase, "A new study published in the journal Science...," you won't hear an equivalent statement mentioning an AERA journal, despite education getting plenty of attention from NBC.

Think tanks have an advantage because the shared ownership of the creation and publication of research creates a common incentive for promotion. Even if the research is lower quality, the spread of the research to a wide audience gives the research power and influence. The traditional system of university-based researchers transferring rights to publishers in exchange for publication might produce higher-quality work, but leaves us with a publication paradox: how do creators promote something they don't own, and how do owners promote something they did not create?

I see two options for improving the incentives to promote academic research: (a) publishers should own creation, or (b) creators should maintain ownership (or at least rights to open distribution). Option (a) essentially turns a publisher into a think tank, and would not fit with academia's culture of academic freedom and independence. Some universities host their own journals, but they do not do so for the purpose of sponsoring and publishing their own work. Furthermore, most university researchers don't want their work to be seen as "work for hire." Option (b), which is not without its challenges, is the better option, and the growing Open Access movement is making it a more viable option every day. But for it to be successful, researchers are going to have to support change -- not for selfish reasons, and not out of spite for publishers, but to ensure the best research is freely available to the audience for which it was intended.

References



Yettick, H. (2009). The research that reaches the public: Who produces the educational research mentioned in the news media? (p. 37). Boulder and Tempe: Education and the Public Inerest Center & Education Policy Research Unit. Retrieved from http://nepc.colorado.edu/publication/research-that-reaches

Yettick, H. (2011, May). Media, think tanks, and educational research. Academe Online. Washington, D.C. Retrieved from http://www.aaup.org/AAUP/pubsres/academe/2011/MJ/Feat/Yett.htm