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: Validity and Causal Inference

With the start of my comprehensive exams beginning in 12 days, my studying has hit the homestretch. Thankfully, my advisor has inspired some confidence by telling me that my understanding of the math education literature is solid and I won't need any more studying in that area. That's good for my studying, and something I take as a huge compliment. So now I can focus for a while on preparing myself for the exam question Derek Briggs is likely to throw my way. Typically, one of the three people on a comps committee is tasked with asking a question related to either the quantitative or qualitative research methodology we learn in our first year of our doctoral program. Derek is a top-notch quantitative researcher, and I enjoyed taking two classes from him last year: Measurement in Survey Research and Advanced Topics in Measurement. Where this gets slightly tricky is that Derek didn't actually teach either of my first-year quantitative methods courses, so there's a potential I could get surprised by something he normally teaches in those classes that I didn't see. It's a risk I was willing to take after working with Derek more recently and more closely in the two measurement courses last year.

It certainly won't be a surprise if Derek asks a question that focuses on issues of validity and causal inference. He mentioned it to me personally and put it in a study guide, so studying it now will be time well spent. I feel like I've had a tendency to read the validity literature a bit too quickly or superficially, so this is a good opportunity for me to revisit some of the papers I've looked at over the past couple of years. Here's the list I've put together for myself:

AERA/APA/NCME. (1999). Standards for educational and psychological testing. Washington, D.C.: American Educational Research Association. [Just the first chapter, "Validity."]

Angoff, W. H. (1988). Validity: An evolving concept. In H. Wainer & H. Braun (Eds.), Test validity (pp. 19–32). Mahwah, NJ: Lawrence Erlbaum Associates.

Borsboom, D., Cramer, A. O. J., Kievit, R. A., Scholten, A. Z., & Franic, S. (2009). The end of construct validity. In R. W. Lissitz (Ed.), The concept of validity: Revisions, new directions, and applications (pp. 135–170). Information Age Publishing.

Brookhart, S. M. (2003). Developing measurement theory for classroom assessment purposes and uses. Educational Measurement: Issues and Practice, 22(4), 5–12. doi:10.1111/j.1745-3992.2003.tb00139.x

Chatterji, M. (2003). Designing and using tools for educational assessment (p. 512). Boston, MA: Allyn & Bacon. [Chapter 3, "Quality of Assessment Results: Validity, Reliability, and Utility"]

Cronbach, L. J. (1988). Five perspectives on validity argument. In H. Wainer & H. I. Braun (Eds.), Test validity (pp. 3–17). Hillsdale, NJ: Lawrence Erlbaum.

Eisenhart, M. A., & Howe, K. R. (1992). Validity in educational research. In M. LeCompte, W. Milroy, & J. Priessle (Eds.), The handbook of qualitative research in education (pp. 642–680). San Diego, CA: Academic Press.

Gorin, J. S. (2007). Test design with cognition in mind. Educational Measurement: Issues and Practice, 25(4), 21–35. doi:10.1111/j.1745-3992.2006.00076.x

Haertel, E. H., & Herman, J. L. (2005). A historical perspective on validity arguments for accountability testing. In J. L. Herman & E. H. Haertel (Eds.), Uses and misuses of data for educational accountability and improvement (NSSE 104th., pp. 1–34). Malden, MA: Wiley-Blackwell.

Holland, P. W. (1986). Statistics and causal inference. Journal of the American Statistical Association, 81(396), 945–960. doi:10.2307/2289069

Kane, M. T. (1992). An argument-based approach to validity. Psychological Bulletin, 112(3), 527–535. doi:10.1037/0033-2909.112.3.527

Leighton, J. P., & Gierl, M. J. (2004). Defining and evaluating models of cognition used in educational measurement to make inferences about examinees’ thinking processes. Educational Measurement: Issues and Practice, 26(2), 3–16. doi:10.1111/j.1745-3992.2007.00090.x

Linn, R. L., & Baker, E. L. (1996). Can performance-based student assessments be psychometrically sound? Performance-based student assessment: Challenges and possibilities (pp. 84–103). Chicago, IL: The University of Chicago Press.

Messick, S. (1988). The once and future issues of validity: Assessing the meaning and consequences of measurement. In H. Wainer & H. I. Braun (Eds.), Test validity (pp. 33–45). Hillsdale, NJ: Lawrence Erlbaum.

Michell, J. (2009). Invalidity in validity. In R. W. Lissitz (Ed.), The concept of validity: Revisions, new directions, and applications (pp. 111–133). Information Age Publishing.

Shadish, W. R., Cook, T. D., & Campbell, D. T. (2002). Experimental and quasi-experimental designs for generalized causal inference (p. 623). Boston, MA: Houghton Mifflin. [Probably Chapters 1-3 and 11, if not more.]

Shepard, L. A. (1993). Evaluating test validity. Review of Research in Education, 19(1), 405–450.

Shepard, L. A. (1997). The centrality of test use and consequences for test validity. Educational Measurement: Issues and Practice, 16(2), 5–24. doi:10.1111/j.1745-3992.1997.tb00585.x

Zumbo, B. D. (2009). Validity as contextualized and pragmatic explanation, and its implications for validation practice. In R. W. Lissitz (Ed.), The concept of validity: Revisions, new directions, and applications (pp. 65–82). Information Age Publishing.

Thankfully, some of these papers I've read recently for my Advances in Assessment course so the amount of reading I have to do is appreciably less than it might look. In my typical fashion, I'll study these in chronological order with the hopes that I get a sense for how the field has evolved its thinking and practice regarding these ideas over the past several decades.

Although I have little other graduate school experience to compare it to, I feel like this reading list is representative of what sets a PhD apart, particularly one earned at an R1 university. It's not necessarily glamorous, and its relevance to the day-to-day teaching and learning in classrooms might not be immediately obvious. But without attending to issues like validity and causal inference, we have a much more difficult time being sure about what we know and how we're using that knowledge. Issues of validity should be at the heart of any assessment or measurement, and when they're attended to properly we greatly improve our ability to advance educational theories and practice.

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.

RYSK: Ball, Thames, & Phelps's Content Knowledge for Teaching: What Makes It Special? (2008)

This is the 17th in a series describing "Research You Should Know" (RYSK) and part of my OpenComps. I also Storified this article as I read.

My last two posts summarized the underpinnings of Shulman's pedagogical content knowledge and Deborah Ball's early work building upon and extending Shulman's theories. Now we jump from Ball's 1988 article to one she co-authored in 2008 with University of Michigan colleagues Mark Thames and Geoffrey Phelps, titled Content Knowledge for Teaching: What Makes It Special?

This article starts by looking at the 20+ years we've had to further develop Shulman's theories of pedagogical content knowledge (PCK). Despite the theory's widespread use, Ball and colleagues claim it "has lacked definition and empirical foundation, limiting its usefulness" (p. 389). (See also Bud Talbot's 2010 blog post and related efforts.) In fact, the authors found that a third of the more than 1200 articles citing Shulman's PCK

do so without direct attention to a specific content area, instead making general claims about teacher knowledge, teacher education, or policy. Scholars have used the concept of pedagogical content knowledge as though its theoretical founcations, conceptual distinctions, and empirical testing were already well defined and universally understood. (p. 394)

To build the empirical foundation that PCK needs, Ball and her research team did a careful qualitative analysis of data that documented an entire year of teaching (including video, student work, lesson plans, notes, and reflections) for several third grade teachers. Combined with their own expertise and experience, and other tools for examining mathematical and pedagogical perspectives, the authors set out to bolster PCK from the ground up:

Hence, we decided to focus on the work of teaching. What do teachers need to do in teaching mathematics -- by virtue of being responsible for the teaching and learning of content -- and how does this work demand mathematical reasoning, insight, understanding, and skill? Instead of starting with the curriculum, or with standards for student learning, we study the work that teaching entails. In other words, although we examine particular teachers and students at given moments in time, our focus is on what this actual instruction suggests for a detailed job description. (p. 395)

For Ball et al., this includes everything from lesson planning, grading, communicating with parents, and dealing with administration. With all this information, the authors are able to sharpen Shulman's PCK into more clearly defined (and in some cases, new) "Domains of Mathematical Knowledge for Teaching." Under subject matter knowledge, the authors identify three domains:

• Common content knowledge (CCK)
• Specialized content knowledge (SCK)
• Horizon content knowledge

And under pedagogical content knowledge, the authors identify three more domains:

• Knowledge of content and students (KCS)
• Knowledge of content and teaching (KCT)
• Knowledge of content and curriculum

Ball describes each domain and uses some examples to illustrate, mostly from arithmetic. For my explanation, I'll instead use something from high school algebra and describe how each domain applied to my growth of knowledge over my teaching career.

Common Content Knowledge (CCK)

Ball et al. describe CCK as the subject-specific knowledge needed to solve mathematics problems. The reason it's called "common" is because this knowledge is not specific to teaching -- non-teachers are likely to have it and use it. Obviously, this knowledge is critical for a teacher, because it's awfully difficult and inefficient to try to teach what you don't know yourself. As an example of CCK, my knowledge includes the understanding that \((x + y)^2 = x^2 + 2xy + y^2\). I've known this since high school, and I would have known it whether or not I became a math teacher.

Specialized Content Knowledge (SCK)

SCK is described by Ball et al. as "mathematical knowledge and skill unique to teaching" (p. 400). Not only do teachers need this knowledge to teach effectively, but it's probably not needed for any other purpose. For my example, I need to have a specialized understanding of how \((x+y)^2\) can be expanded using FOIL or modeled geometricaly with a square. It may not be all that important for students to understand both the algebraic and geometric ways of representing this problem, but I need to know both so I can better understand student strategies and sources of error. Namely, the error that \((x + y)^2 = x^2 + y^2\).

Horizon Content Knowledge

This domain was provisionally included by the authors and described as, "an awareness of how mathematical topics are related over the span of mathematics included in the curriculum" (p. 403). For my example of \((x + y)^2 = x^2 + 2xy + y^2\), I need to understand how previous topics like order of operations, exponents, and the distributive property relate to this problem. Looking forward, I need to understand how this problem relates to factoring polynomials and working with rational expressions.

Knowledge of Content and Students (KCS)

This is "knowledge that combines knowing about students and knowing about mathematics" (p. 401) and helps teachers predict student thinking. KCS is what allows me to expect students to incorrectly think \((x + y)^2 = x^2 + y^2\), and to tie that to misconceptions about the distributive property and exponents. I'm not sure I had this knowledge for this example when I started teaching, but it didn't take me long to figure out that it was a very common student mistake.

Knowledge of Content and Teaching (KCT)

Ball et al. say KCT "combines knowing about teaching and knowing about mathematics" (p. 401). While KCS gave me insight about why students mistakingly think \((x + y)^2 = x^2 + y^2\), KCT is the knowledge that allows me to decide what to do about it. For me, this meant choosing a geometric representation for instruction over using FOIL, which lacks the geometric representation and does little to address the problem if students never recognize that \((x + y)^2 = (x + y)(x + y)\).

Knowledge of Content and Curriculum

For some reason, Ball et al. include this domain in a figure in their paper but never describe it explicitly. They do, however, scatter enough comments about knowledge of content and curriculum to imply that teachers need a knowledge of the available materials they can use to support student learning. For my example, I know that CPM uses a geometric model for multiplying binomials, Algebra Tiles/Models can be used to support that model, virtual tiles are available at the National Library of Virtual Manipulatives (NLVM), and the Freudenthal Institute has an applet that allows students to interact with different combinations of constants and variables when multiplying polynomials.

Some of the above can be hard to distinguish, but thankfully Ball and colleagues clarify by saying:

In other words, recognizing a wrong answer is common content knowledge (CCK), whereas sizing up the nature of an error, especially an unfamiliar error, typically requires nimbleness in thinking about numbers, attention to patterns, and flexible thinking about meaning in ways that are distinctive of specialized content knowledge (SCK). In contrast, familiarity with common errors and deciding which of several errors students are most likely to make are examples of knowledge of content and students (KCS). (p. 401)

In their conclusion, the authors hope that this theory can better fill the gap that teachers know is important, but isn't purely about content and isn't purely about teaching. We can hope to better understand how each type of knowledge above impacts student achievement, and optimize our teacher preparation programs to reflect that understanding. Furthermore, that understanding could be used to create new and improved teaching materials and professional development, and better understand what it takes to be an effective teacher. With this in mind, you can gain some insight to what Ball was thinking when she gave this congressional testimony:

References

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

RYSK: Ball's Unlearning to Teach Mathematics (1988)

This is the 16th in a series describing "Research You Should Know" (RYSK) and part of my OpenComps. I also Storified this article as I read.

Dan Lortie's 1975 book Schoolteacher clarified an idea that teachers already know: how we teach is greatly influenced by the way we've been taught. Lortie called the idea apprenticeship of observation, and it specifically refers to how teachers, having spent 13,000+ hours in classrooms as students, take that experience as a lesson in how to be a teacher. What we often fail to deeply reflect on, however, is that we were only seeing the end product of teaching. We didn't see the lesson planning, go to summer conferences, attend professional development workshops, study the science of learning, or take part in the hundreds of decisions a teacher makes every day. Just observing isn't a proper apprenticeship, even after thousands of hours watching good teachers. I think of it this way: I watch a lot of baseball, and I can tell good baseball from bad. This hardly makes me ready to play, sadly, because I'm not spending hours taking batting practice, participating in fielding drills, studying video, digesting scouting reports, and working out in the offseason. Just as watching a lot of baseball doesn't really prepare me to play baseball, watching a lot of teaching doesn't really prepare someone to teach. Still, all those hours heavily influence our beliefs, both of teaching and of subject matter.

Deborah Ball (CC BY-NC-ND
House Committee on Education
and the Workforce Democrats)

In 1988, the year she earned her Ph.D at Michigan State, Deborah Ball was spending a lot of time thinking about math teachers' apprenticeship of observation. In an article called Unlearning to Teach Mathematics, she describes a project involving teaching permutations to her class of introductory preservice elementary teachers. The goal was not simply to teach her students about permutations, but also to learn more about their beliefs about the nature of mathematics and to develop strategies that might enlighten those beliefs and break the cycle of simply teaching how you were taught.

By selecting permutations as the topic, Ball hoped to expose these introductory teachers to a topic they'd never studied formally. By carefully observing how her students constructed their knowledge, Ball would be able to see how their prior understandings about mathematics influenced their learning. The unit lasted two weeks. In the first phase of the unit, Ball tried to engage the students in the sheer size and scope of permutations, like by thinking about how the 25 students could be sat in 1,551,121,000,000,000,000,000,000 different seating arrangements. Working back to the simplest cases, with 2, 3, and 4, students, students could think and talk about the patterns that emerge and understand how the permutation grows so quickly. For homework, Ball asked students to address two goals: increase their understanding of permutations, but also think about the role homework plays in their learning, including how they approach and feel about it and why. In the second phase of the unit, Ball has her students observe her teaching young children about permutations, paying attention to the teacher-student interactions, the selection of tasks, and what the child appears to be thinking. In the last phase of the unit, the students become teachers and try helping someone else explore the concept of permutations. After discussing this experience, students wrote a paper reflecting on the entire unit.

From other research, Ball knew that teacher educators often assumed their students had mastery of content knowledge. Even moreso, future elementary math teachers themselves assumed they had mastery over the mathematical content they'd be expected to teach. She knew, however, that there was something extra a teacher needed to teach that content. Citing Shulman's pedagogical content knowledge, along with numerous others, Ball describes some ways we can think about what that special content knowledge for teaching is, but admits that her permutations project was too narrow to explore how teachers construct and organzie that knowledge. The project would, however, give insight to her students' ideas about mathematics, and assumptions they make about what it means to know mathematics. For example, a student named Cindy wrote:

I have always been a good math student so not understanding this concept was very frustrating to me. One thing I realized was that in high school we never learned the theories behind our arithmetic. We just used the formulas and carried out the problem solving. For instance, the way I learned permutations was just to use the factorial of the number and carry out the multiplication ... We never had to learn the concepts, we just did the problems with a formula. If you are only multiplying to get the answer every time, permutations could appear to be very easy. If you ask yourself why do we multiply and really try to understand the concept, then it may be very confusing as it was to me. (p. 44)

Comments like this revealed that many of Ball's students relied on a procedural view of mathematics, one where the question "Why?" had been rarely asked. Ball also noticed a theme in her students' reflections about knowing math "for yourself" versus for teaching. Alison wrote:

I was trying to teach my mother permutations. But it turned out to be a disaster. I understood permutations enough for myself, but when it came time to teach it, I realized that I didn't understand it as well as I thought I did. Mom asked me questions I couldn't answer. Like the question about there being four times and four positions and why it wouldn't be 4 x 4 = 16. She threw me with that one and I think we lost it for good there.

From observing a young student learn about permutations in phase two, Ball noticed that some of her students started to challenge some of their assumptions they made about themselves as learners. Both from her experience and from the literature, Ball knew that elementary preservice teachers are often the most apprehensive about teaching mathematics. In some cases, these students choose to teach elementary in the hopes of avoiding any mathematical content they might find difficult. Changing these feelings about mathematics and about themselves is a difficult task for the teacher educator, but Ball did see progress. Christy, for example, said, "Most of all, I realized that I do have the ability to learn mathematics when it is taught in a thoughtful way" (p. 45). Unfortunately, not all shared this experience, as Mandy said she "did not enjoy the permutations activities because I was transported in time back to junior high school, where I remember mathematics as confusing and aggravating. Then as now, the explanations seemed to fly by me in a whirl of disassociated numbers and words" (p. 45).

In her conclusion, Ball says activities like the permutations project can be used by teacher educators to expose students' "knowledge, beliefs, and attitudes" (p. 46) about math and teaching math. By understanding the ideas prospective teachers bring with them, teacher educators can better develop preparation programs that address those beliefs in ways that strengthen the positive ones while changing some negative ones. Also, by including these kinds of activities with introductory preservice teachers, this can raise their expectations for what they will encounter later in methods classes. Summarizing, Ball concludes:

How can teacher educators productively challenge, change, and extend what teacher education students bring? Knowing more about what teachers bring and what they learn from different components of and approaches to professional preparation is one more critical piece to the puzzle of improving the impact of mathematics teacher education on what goes on in elementary mathematics classrooms. (p. 46)

References

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

RYSK: Shulman's Those Who Understand: Knowledge Growth in Teaching (1986)

This is the 15th in a series describing "Research You Should Know" (RYSK) and part of my OpenComps. I also Storified this article as I read.

Lee Shulman. (CC BY-NC) Penn State

George Bernard Shaw once said, "He who can, does. He who cannot, teaches." For that, you could say that Lee Shulman takes offense. Shulman, a long-time faculty member at both Michigan State (1963-1982) then Stanford, explained his position and a new way of thinking about teacher knowledge in his AERA Presidential Address and the paper, Those Who Understand: Knowledge Growth in Teaching. Shulman is now an emertius professor but stays active traveling, speaking, and occasionally blogging.

Wondering why the public often has a low opinion of teachers' knowledge and skill, Shulman first looks at the history of teacher examinations. In the latter half of the 1800s, examinations for people wishing to teach were almost entirely content-based. In 1875, for example, the California State Board examination for elementary teachers gave a day-long, 1000-point exam that covered everything from mental arithmetic to geography to vocal music. Its section on the theory and practice of teaching, however, was only worth 50 of the 1000 points and included questions like, "How do you interest lazy and careless pupils?" (p. 5)

By the 1980s, when Shulman wrote this article, teacher examinations painted almost the opposite picture. Instead of focusing on content, they focused on topics such as lesson planning, cultural awareness, and other aspects of teacher behavior. While the topics usually had roots in research, they clearly did not represent the wide spectrum of skills and knowledge a teacher would need to be a successful teacher. More specifically, by the 1980s our teacher examinations seemed to care as little about content as the examinations a century prior seemed to care about pedagogy.

Looking back even further in history, Shulman recognized that we haven't always made this distinction between content and teaching knowledge. The origins of the names of our highest degrees, "master" and "doctor," both essentially mean "teacher" and reflected the belief the highest form of knowing was teaching, an idea going back to at least Aristotle:

We regard master-craftsmen as superior not merely because they have a grasp of theory and know the reasons for acting as they do. Broadly speaking, what distinguishes the man who knows from the ignorant man is an ability to teach, and this is why we hold that art and not experience has the character of genuine knowledge (episteme) -- namely, that artists can teach and others (i.e., those who have not acquired an art by study but have merely picked up some skill empirically) cannot. (Wheelwright, 1951, as cited in Shulman, 1986, p. 7)

Shulman saw a blind spot in this dichotomy between content and teaching knowledge. What he saw was a special kind of knowledge that allows teachers to teach effectively. After studying secondary teachers across subject areas, Shulman and his fellow researchers looked to better understand the source of teachers' comprehension of their subject areas, how that knowledge grows, and how teachers understand and react to curriculum, reshaping it into something their students will understand.

Pedagogical Content Knowledge

To better understand this special knowledge of teaching, Shulman suggested we distinguish three different kinds of content knowledge: (a) subject matter knowledge, (b) pedagogical content knowledge, and (c) curricular knowledge. It was the second of these, pedagogical content knowledge (PCK), that Shulman is best remembered for. Shulman describes the essence of PCK:

Within the category of pedagogical content knowledge I include, for the most reguarly taught topics in one's subject area, the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations -- in a word, the ways of representing and formulating the subject that make it comprehensible to others. Since there are no single most powerful forms of representation, the teacher must have at hand a veritable armamentarium of alternative forms of representation, some of which derive from research whereas others originate in the wisdom of practice. (p. 9)

In addition to these three kinds of teacher knowledge, Shulman also proposed we consider three forms of teacher knowledge: (a) propositional knowledge, (b) case knowledge, and (c) strategic knowledge. These are not separate from the three kinds of knowledge named above, but rather describe different forms of each kind of teacher knowledge. Propositional knowledge consists of those things we propose teachers do, from "planning five-step lesson plans, never smiling until Christmas, and organizing three reading groups" (p. 10). Shulman organized propositional knowledge into principles, maxims, and norms, with the first usually emerging from research, the second coming from a practical experience (and generally untestable, like the suggestion to not smile before Christmas), and the third concerning things like equity and fairness. Propositions can be helpful but difficult to remember to implement as research intended.

Learning propositions out of context is difficult, so Shulman proposed case knowledge as the second form of teacher knowledge. By case, he means learning about teaching in a similar way a lawyer learns about the law by studying prior legal cases. In order to truly understand a case, a learner starts with the factual information and works towards the theoretical aspects that explain why things happened. By studying well-documented cases of teaching and learning, teachers consider prototype cases (that exemplify the theoretical), precedents (that communicate maxims), and parables (that communicate norms and values). (If you're scoring at home, Shulman has now said there are three types of cases, which itself is one of three forms of knowledge, each of which capable of describing three different kinds of content knowledge.)

The last form of knowledge, strategic knowledge, describes how a teacher reacts when faced with contradictions of other knowledge or wisdom. Knowing when to bend the rules or go against conventional wisdom takes more than luck -- it requires a teacher to be "not only a master of procedure but also of content and rationale, and capable of explaining why something is done" (p. 13).

The value of this article by Shulman goes beyond the theoretical description of pedagogical content knowledge. Additionally, this article serves as a strong reminder that when we judge a teacher, we must consider a broad spectrum of skills and abilities, and not limit ourselves to only those things we think can be easily measured. As Shulman explains:

Reinforcement and conditioning guarantee behavior, and training produces predictable outcomes; knowledge guarantees only freedom, only the flexibility to judge, to weigh alternatives, to reason about both ends and means, and then to act while reflecting upon one's actions. Knowledge guarantees only grounded unpredictability, the exercise of reasoned judgment rather than the display of correct behavior. If this vision constitutes a serious challenge to those who would evaluate teaching using fixed behavioral criteria (e.g., the five-step lesson plan), so much the worse for those evaluators. The vision I hold of teaching and teacher education is a vision of professionals who are capable not only of acting, but of enacting -- of acting in a manner that is self-conscious with repect to what their act is a case of, or to what their act entails. (p. 13)

In our current era of teacher evaluation and accountability, with all its observational protocols and test score-driven value added models, this larger view of teaching presented to us by Shulman is a gift. His recommendation that teacher evaluation and examination "be defined and controlled by members of the profession, not by legislators or laypersons" (p. 13) is a wise one, no matter how politically difficult. Shulman hoped for tests of pedagogical content knowledge that truly measured those speical skills that teachers have, skills that non-teaching content experts would not pass. I don't think those measurement challenges have been overcome, but continuing towards that goal should strengthen teacher education programs while also improving the perception of teaching as a profession. As Shulman concludes (p. 14):

We reject Mr. Shaw and his calumny. With Aristotle we declare that the ultimate test of understanding rests on the ability to transform one's knowledge into teaching.

Those who can, do. Those who understand, teach.

References

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

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: Gravemeijer's Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education (2004)

This is the 12th in a series describing "Research You Should Know" (RYSK) and part of my OpenComps.

Gravemeijer (from above) at the 2011 RME Conference

I began my recent reading of the literature on learning trajectories by reading Clements & Sarama's (2004) Learning Trajectories in Mathematics Education, and then went back to where the idea formally began, Simon's (1995) Reconstructing Mathematics Pedagogy from a Constructivist Perspective. Now I'm jumping to 2004 again with Koeno Gravemeijer's Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education. Koeno Gravemeijer (pronounced Koo-no Grav-meyer) has worked at multiple institutions in the Netherlands and spent time at Vanderbilt working with Paul Cobb, but he's best known for his long time association and leadership with the Freudenthal Institute and his advancements of Realistic Mathematics Education (RME).

When Martin Simon introduced the concept of hypothetical learning trajectories in his 1995 paper Reconstructing Mathematics Pedagogy from a Constructivist Perspective, he described them as part of a teaching cycle that was informed by the teacher's knowledge and then revised after assessment of student understanding. While much of the focus was placed on the idea of the trajectory, Simon made clear that no two trajectories will be alike, as each one is hypothesized for a unique group of students who are uniquely constructing knowledge. In other words, you can't just prescribe a trajectory and ask teachers to follow it to the letter. Instead, Simon suggested we needed to build an understanding of the knowledge teachers were using to inform and modify their trajectories:

A possible contribution that can be made by the analysis of data and the resulting model reported in this paper is to encourage other researchers to examine teachers' "theorems in action" and to make teachers' assumptions, beliefs, and emerging theories about teaching explicit. (p. 142)

This paper by Gravemeijer is, in part, a response to Simon's call to other researchers. Gravemeijer first states that in a constructivism-inspired reform mathematics, the traditional goals of instructional design must change:

What is needed for reform mathematics education is a form of instructional design supporting instruction that helps students to develop their current ways of reasoning into more sophisticated ways of mathematical reasoning. For the instructional designer this implies a change in perspective from decomposing ready-made expert knowledge as the starting point for design to imagining students elaborating, refining, and adjusting their current ways of knowing. (p. 106)

Next, Gravemeijer recognizes that while every teacher can use their knowledge to hypothesize a learning trajectory, we (researchers, teacher educators, curriculum designers) need to have some knowledge in common if we want to help teachers:

The example Simon (1995) worked out shows that designing hypothetical learning trajectories for reform mathematics is no easy task. We can, therefore, ask ourselves what kind of support can be given to teachers. It is clear that we cannot rely on fixed, ready-made, instructional sequences, because the teacher will continuously have to adapt to the actual thinking and learning of his or her students. Thus it seems more adequate to offer the teacher some framework of reference, and a set of exemplary instructional activities that can be used as a source of inspiration. (p. 107)

This is where Gravemeijer introduces the concept of a local instruction theory, which he describes as "the description of, and rationale for, the envisioned learning route as it relates to a set of instructional activities for a specific topic" (p. 107). I admit, it's difficult at first to discern this from a hypothetical learning trajectory, but I think the key is the relationship to instructional activities (which are more fixed/solid) instead of a trajectory's relationship to student understanding (which is more flexibile/fluid). By addressing the relationship of learning to the instructional activities, Gravemeijer uses local instruction theories to describe a common foundation teachers can use for building trajectories, saying that "Externally developed local instruction theories are indispensable for reform mathematics education" and that it is "unfair to expect teachers to invent hypothetical learning trajectories without any means of support" (p. 108). (If you're still confused, I think I can safely oversimplify it like this: Simon says trajectories are about student learning, not mathematical tasks. Gravemeijer agrees, but since trajectories are unique because student learning is unique, it helps if we have some agreed-upon ideas about how mathematical tasks should be designed.) Given Gravemeijer's long association with the Freudenthal Institute, he naturally describes how design principles from Realistic Mathematics Education (RME) provide the kind of instructional design framework for creating a local instruction theory.

Design Research and RME

Some curricula and instructional strategies are developed then subjected to treatment and control groups to test their effectiveness. That's not design research and not how RME has been developed. Instead, design research consists of cyclical iterations of thought experiments, teaching experiments, and retrospective analyses. It's similar to how teachers improve their instruction as they gain experience: they plan an activity for year one, then conduct that activity, then reflect on the activity so it will be better in year two. Of course, a team of researchers who are carefully theorizing, observing, collecting data, and analyzing the results across multiple classrooms can more quickly and effectively improve tasks and instruction than a teacher can alone.

Gravemeijer describes the design research he conducted with Paul Cobb and others around the development of mental computation strategies for addition and subtraction with elementary students. There are numerous papers and at least part of one dissertation all related to this work, so I won't describe it here. I will, however, describe the three RME design principles that Gravemeijer cites as helping form the local instruction theory that guided the design research process.

Guided Reinvention

Hans Freudenthal (1973) believed mathematics is best learned when students get to experience a process of learning that's similar to the way the mathematics was invented.

If mathematics is to be applied, applying mathematics should be taught and learned. Applying is often interpreted, as mentioned above, as substituting numerical values for parameters in general theorems and theories. This is a misleading terminology. Mathematics is applied by creating it anew each time -- I will expound this in more detail too. This activity can never be exercised by learning mathematics as a ready-made product. Drilling algorithms may be indispensable, but inventing problems to drill algorithms does not create opportunities to teach applying mathematics. This so-called applied mathematics lacks the flexibility of good mathematics. (Freudenthal, 1973, p. 118)

I've heard criticisms of this approach. "How in the world can a student reinvent mathematics that took mathematicians hundreds of years to understand?" That's a valid question, and the best answer is: "Through carefully designed curriculum and instruction." The goal is not to replicate the invention of the mathematics, but learn from history how a mathematical idea might be constructed in the mind of a student. Of course, this takes an extensive and special knowledge of the history of mathematics, and largely explains why Freudenthal's Mathematics as an Educational Task is almost 700 pages long.

Didactical Phenomenology

The concept of didactical phenomenology relates the mathematical "thought thing" and the phenomenon it describes. This is not a theory I know well but hope to study more in the future.

Mathematical concepts, structures, and ideas serve to organise phenomena -- phenomena from the concrete world as well as from mathematics -- and in the past I have illustrated this by many examples. By means of geometrical figures like triangle, parallelogram, rhombus, or square, one succeeds in organising the world of contour phenomena; numbers organise the phenomenon of quantity. On a higher level the phenomenon of geometrical figure is organised by means of geometrical constructions and proofs, the phenomenon "number" is organised by means of the decimal system. So it goes in mathematics up to the highest levels: continuing abstraction brings similar looking mathematical phenomena under one concept -- group, field, topological space, deduction, induction, and so on. (Freudenthal, 1983, p. 28)

Traditionally we teach an abstract mathematics and then find examples for students to make the mathematics concrete. With didactical phenomenology, we focus on progressive mathematization, suggesting "looking for phenomena that might create opportunities for the learner to constitute the mental object that is being mathematized" (Gravemeijer, p. 116). Yes, it's hard to understand without a lot of specific examples, and that's why Freudenthal wrote almost 600 pages on this topic. It's all in a book I have yet to read, so I'll forgive myself for giving a better description here.

Emergent Modeling

I can best describe emergent modeling with an example. Imagine an elementary class learning about fractions. Instead of giving students a formal model (like a numerator and denominator), the concept of emergent modeling says we should let students reach these models informally and progressively. If a task involves the sharing of parts of cookies with the students, students might begin with breaking apart actual cookies. Once realizing this isn't convenient, students might move to drawing cookies on paper. At some point they'll realize that drawing all the details of the cookie isn't necessary and just use a circle to represent a cookie. Up until this point, these are all models-of a cookie. The key step in this process is when students start using circles to model other contextual situations, like working with fractions of time, money, space, etc. Now the circle is a model-for a part-whole relationship, and not representing a specific object like a cookie. These models-for have the power to generalize to other contexts, and eventually students no longer need the circle and rely on formal mathematics to represent and work with fractions. Gravemeijer describes a similar process in this paper, except with how bead strings, unifix cubes, and rulers can lead to marked and empty number lines as students develop ideas of cardinality, ordinality, and distance as they learn mental strategies for addition and subtraction.

Conclusion

I hope by now you have some sense for a local instruction theory. The three RME principles above -- guided reinvention, didactical phenomenology, and emergent modeling -- do not describe a detailed instructional sequence of tasks and instructions for a teacher. They are, however, a way of theorizing how a particular instructional sequence should work, grounded in the design research conducted by Gravemeijer et al. This kind of local instruction theory is what allows teachers to design hypothetical learning trajectories that focus on the construction of student understanding, and provide some common ground for helping teachers become better at trajectory hypothesizing.

References

Freudenthal, H. (1973). Mathematics as an educational task (p. 680). Dordrecht, The Netherlands: D. Reidel.

Freudenthal, H. (1983). Didactical phenomenology of mathematical structures (p. 595). Dordrecht, The Netherlands: D. Reidel.

Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128. doi:10.1207/s15327833mtl0602_3

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. doi:10.2307/749205

RYSK: Simon's Reconstructing Mathematics Pedagogy from a Constructivist Perspective (1995)

This is the 11th in a series describing "Research You Should Know" (RYSK) and part of my OpenComps.

After reading Clements & Sarama's (2004) Learning Trajectories in Mathematics Education a few days ago, I wanted to go back to the origins of learning trajectories: a 1995 paper from Martin Simon that explored how mathematics can and should be taught differently with a constructivist mindset. Simon is a professor of math education at NYU and has a history of researching how students and their teachers come to understand their mathematical knowledge.

From the start, I immediately appreciated two strengths of this article: Simon's clear writing and the relatively straightforward description of constructivism he offers. When you're still trying to sort out what constructivism is and is not (like me and many classroom teachers), it's a whole lot easier to parse this article from 1995 than, say, a heavily theoretical, mid-2000s piece by Jim Greeno. Recognizing that there are multiple (and often subtly different) ways to describe constructivism, Simon lays out his interpretation like this:

Constructivism derives from a philosophical position that we as human beings have no access to an objective reality, that is, a reality independent of our way of knowing it. Rather, we construct our knowledge of our world from our perceptions and experiences, which are themselves mediated through our previous knowlege. Learning is the process by which human beings adapt to their experiential world. (p. 115)

So when we have an idea that "works," meaning it does what we need it to do to make sense of our experiences, then we've constructed knowledge. This can be a tough sell to the mathematically-minded who say things like, "I didn't construct two plus two equals four. That's an objective fact." In response, the constructivist would disagree about having access to objective reality, but acknowledge that we (almost?) universally construct the knowledge that 2+2=4 because we experience no evidence suggesting otherwise (what Simon and others refer to as disequilibrium).

There's also a theoretical debate about the construction of knowledge as an individual, cognitive process versus a social process. This is an interesting debate, to be sure, and it has been pushing the leading edges of theories for learning mathematics for about the past twenty years. If you're wearing a theoretician hat, then you care deeply about how this debate might be won. But if you're wearing a researcher hat (like Simon is here), then you use both theories to help you gain whatever insights they might afford you. Simon (crediting work by Cobb, Yackel, and Wood) calls this coordination of psychological and sociological approaches "social constructivism," and compares it to how a physicist can better explain the nature of light by considering it both a particle and a wave.

Simon takes pains in this article to separate the theory of constructivism from a notion of "constructivist teaching." It's a mistake that I've seen and heard many times, and it's important to understand the differences and nuances. Simon states:

As I stated above, constructivism, as an epistemological theory, does not define a particular way of teaching. It describes knowledge development whether or not there is a teacher present or teaching is going on. ... There is no simple function that maps teaching methodology onto constructivist principles. A constructivist epistemology does not determine the appropriateness or inappropriateness of teaching strategies. ... The commonly used misnomer, "constructivist teaching," [suggests that] constructivism offers one set notion of how to teach. The question of whether teaching is "constructivist" is not a useful one and diverts attention from the more important question of how effective it is. From a theoretical perspective, the question that needs attention is, In what ways can constructivism contribute to the development of useful theoretical frameworks for mathematics pedagogy? (p. 117)

Using this perspective and a lot of theoretical support from work done in the early 1990s, Simon sets out to explore "the ongoing and inherent challenge to integrate the teacher's goals and direction for learning with the trajectory of students' mathematical thinking and learning" (p. 121, emphasis in original). Unlike a traditional perspective, where the pedagogical focus tended towards chopping mathematical content into manageable pieces to be demonstrated and practiced, Simon wished to focus on student understanding and a plan for mathematical tasks that improved that understanding.

I won't describe Simon's teaching experiment in great detail (after all, it was data-rich enough for Simon to publish multiple papers), but it involved a group of preservice elementary teachers and a set of tasks designed to elicit understandings about how multiplication related to the simple area formula A = l x w. Simon knew that his students had no trouble multiplying or using the formula. That wasn't the problem. Instead, he gave them this task:

Determine how many rectangles, of the size and shape of the rectangle that you were given, could fit on the top surface of your table. Rectangles cannot be overlapped, cannot be cut, nor can they overlap the edges of the table. Be prepared to describe to the class how you solved this problem. (p. 123)

Students used their rectangle (I'm imagining an index card) to measure the length and width of their table. A few groups questioned whether the rectangle should maintain its orientation, or if the long edge should always align with the edge of the table. This launched a class discussion and Simon pushed students to explain how they found the area without defaulting to "I used the formula." Some students talked about rows and columns, some talked about counting rectangles, but comments about "overlapping" rectangles suggested misunderstanding was still apparent. Compounding the problem was that these students were not accustomed to provide this level of justification.

Simon tried varying the task to elicit better student explanations. Some students seemed to get it while others still struggled or remained silent. (The transcript excerpts in Simon's paper are very valuable here, if you can get a copy of it.) Simon began to worry that students were actually misunderstanding things about area, not just how multiplication relates to rectangle area, so he assigned a second task about finding the area of an irregular shape. This was less of a problem for students, so Simon returned to the "turned rectangle" problem and tried another activity with measuring tables, both with rectangular cards and also with sticks. Some students were stuck in their thinking that the area unit must be the size and shape of the card, while others began to see how using the long edge of the card for length and width of the table created new, square units not shaped like the card.

All these classes were observed by researchers who took notes and videotaped the classroom activity. Simon also kept a journal of his reflections after each lesson and planning session. Following the teaching experiment, Simon analyzed his role as the decision maker in the classroom activities. First, he had hypothesized that his students would otherwise be satisfied with knowing and using a formula for area, but had probably never explored why the formula worked. This hypothesis was based on prior experience with similar students, prior research, and pretesting. Simon carefully thought out what he thought would happen in his initial activity, saying this thinking

provides an example of the reflexive relationship between the teacher's design of activities and consideration of the thinking that students might engage in as they participate in those activities. The consideration of the learning goal, the learning activities, and the thinking and learning in which students might engage make up the hypothetical learning trajectory, a key part of the Mathematical Learning Cycle described in the next section. (p. 133)

The "Mathematical Learning Cycle" Simon, in a simplified way, suggests how a teacher's knowledge can be used to create a hypothetical learning trajectory (containing a learning goal, a plan for learning activities, and a hypothesis about the learning process), and how assessment of student knowledge gives the teacher new and better knowledge upon which to refine the hypothetical learning trajectory. (I can't help but wonder if Simon once thought he'd be remembered for "learning cycles," not "learning trajectories.") The trajectory as planned is always hypothetical because it is just the teacher's prediction and the true trajectory cannot be known in advance. Modification of the trajectory happens as the teacher increases his/her knowledge about what students understand, which can be during a planning session between lessons or on-the-fly during a classroom activity. Of course, the more knowledge a teacher has in advance -- about their students, about the mathematical content, and about theories of learning that content -- the better the hypothetical learning trajectory can be. Sometimes we don't have all the information we want, says Simon:

As a teacher, I often do not have a well-developed map of the mathematical conceptual area in which I am engaging my students; that is, I may not have fully articulated for myself (or found in the literature) the specific connections that constitute understanding or the nature of development of understanding in that area. ... Thus, in such cases, my operational definition of understanding is the ability to overcome these particular difficulties; I may not have unpacked the difficulties in order to understand the conceptual issues that are implicated. Thus, even if I do not have a thorough knowledge of what constitutes mathematical understanding in a particular domain, having a rich set of problem situations that challenge students and having knowledge of conceptual difficulties that they typically encounter provide me with an approximation that lets me be reasonably effective in promoting learning in the absence of more elaborated knowledge. (This is not to suggest that the more elaborated understanding would not be more powerful.) (p. 139)

In his summary, Simon reiterates some major themes:

1. Student understanding is prioritized in the design of instruction
2. Teachers learn as students learn
3. Planning instruction includes the creation of a hypothetical learning trajectory
4. Because of #2, teachers need to constantly revise #3

Lastly, Simon emphasizes the challenge of teaching using the methods and example he's described. "Teachers will need access to relevant research on children's mathematical thinking, innovative curriculum materials, and ongoing professional support in order to meet the demands of this role" (pp. 142-143). I plan on summarizing more work on learning trajectories, so hopefully I can provide a little bit of that needed support.

References

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. doi:10.2307/749205

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.

RYSK: Clements & Sarama's Learning Trajectories in Mathematics Education (2004)

This is the tenth in a series describing "Research You Should Know" (RYSK) and part of my OpenComps.

When I shared my comps reading list with my committee, Bill Penuel quickly replied with the suggestion that I read this article about learning trajectories by Doug Clements and Julie Sarama. I'd seen Clements present on this topic at last year's RME conference (which focused on learning trajectories/progressions), and I recognized the paper as something I found last spring too late in the writing of a final paper to really read and process, so I am happy to return to it now.

At their most basic, learning trajectories can be thought of as sequences of tasks and activities aimed at the progressive development of mathematical thinking and skill. This appeals to me because, quite frankly, I'm not all that great at focusing on single mathematical tasks. Even with a great task, I find myself wondering, "Where in the curriculum does this task fit? What should students know and be able to do before attempting it? Once students complete this task, what new thing are they ready for?" You could say I get a bit distracted in an effort to see the big picture, a habit of mine that's not necessarily new. Learning trajectories are one way of thinking about curriculum on a larger scale, and the better I understand them, the more organized my thinking can be.

Clements finds the roots of learning trajectories in a 1995 paper by Martin Simon titled Reconstructing Mathematics Pedagogy from a Constructivist Perspective (which I should also add to my comps reading list, I'm sure). While it's certainly possible to create a learning trajectory thinking only about instructional tasks, Clements & Sarama stress the interconnections between the instructional sequence and the psychological developmental progression of students. As teachers, sometimes we make the mistake of breaking down an instructional sequence according to the structure of the mathematics, which may or may not reflect the ways students will actually construct their mathematical knowledge. To avoid this mistake, Clements & Sarama suggest designing learning trajectories using this three-stage process:

1. Specify a research-based learning model that describes how students construct the mathematical knowledge needed for the trajectory. I think this is a tough task for teachers, both because the specific models in the research are not widely known and understood and because there are surely many areas of mathematics for which specific learning models have not been thoroughly studied.
2. Select key mathematical tasks to promote learning at each level of students' psychological development. Again, it takes the help of research to judge if a task truly targets a certain level of development or not.
3. Complete the hypothetical learning trajectory by sequence the tasks to match the students' developmental progression.

Of course, the completed learning trajectory should (a) take advantage of specific and relevant cultural knowledge and practices of your students and (b) be subjected to repeated revision and refinement. Clements & Sarama do not understate the potential of well-constructed learning trajectories:

The enactment of an effective, complete learning trajectory can actually alter developmental progressions or expectations previously established by psychological studies because it opens up new paths for learning and development. This, of course, reflects the traditional debate between Vygotsky (1934/1986) and Piaget and Szeminska (1952) regarding the priority of development over learning. We believe that learning trajectory research, along with other research corpi, suggests the Vygotskian position that, at least in some domains and some ways, learning and teaching tasks can change the course of development. (p. 84)

Finally, Clements & Sarama make two more recommendations regarding the creation of learning trajectories. First, be sure to think carefully about how a trajectory might work for an individual student (following a more cognitive theoretical approach) and also how it might work for a class, complete with student interactions and classroom discourse (a more sociocultural theoretical approach). Second, recognize that these trajectories are always hypothetical and will work best when teachers take the time to create and re-create them to work best with their students.

References

Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89. doi:10.1207/s15327833mtl0602_1

Piaget, J. & Szeminska, A. (1952). The child's conception of number. London, UK: Routledge and Kegan Paul.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. doi:10.2307/749205

Vygotsky, L. S. (1934/1986). Thought and language. Cambridge, MA: MIT Press.

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!