Wednesday, July 30, 2014

On Major Problems and Grand Challenges, Part 1

Last month the NCTM Research Committee asked its members to help it identify the grand challenges for mathematics education. Grand challenges, said NCTM, (a) are hard yet doable, (b) affect millions of people, (c) need a comprehensive research program, (d) are goal-based with progress we can measure, and (e) capture the public's attention and support. I'm a month too late to contribute to NCTM's survey, and before blogging my thoughts into the wider conversation I thought I should look back at someone else's previous attempt. Maybe I'd gain some perspective on what grand challenges are and how persistent they might be.

Hans Freudenthal (Wikimedia Commons, CC-BY-SA)
In 1980, Hans Freudenthal gave a plenary address at ICME that later turned into an article in Educational Studies in Mathematics titled, Major Problems of Mathematics Education. I've briefly summarized the article on the MathEd Wiki and here I'll note the progress I think we've made on Freudenthal's 11 problems.
  1. Freudenthal believed we "need[ed] more pardigmatic cases, paradigms of diagnosis and prescription, for the benefit of practitioners and as bricks for theory builders" (p. 135). In the case of arithmetic, which was Freudenthal's example, I think Cognitively Guided Instruction (CGI) is very much the kind of thing Hans was looking for.
  2. Freudenthal wanted us to more carefully consider how people learn and observe their learning processes. I think several decades of teachers' awareness of constructivist theories of learning has changed how most people think of learning, and newer work in the area of teacher noticing puts fine points on what teachers notice and why.
  3. How do we design curriculum and instruction around progressive formalization? There is always more to learn, but the Freudenthal Institute in the Netherlands has now worked on this for decades and the frameworks for curriculum design are well-established.
  4. How do we retain and leverage mathematical insight? Freudenthal wrapped this into the conceptual vs. procedural debate, one that's still very much alive. However, I think we have better examples of productive approaches to this problem, and some research results (the BEAR project work at Berkeley comes to mind) showed that more focus on the conceptual didn't come at the expense of procedural facility. Still, this problem gets wrapped up in people's beliefs about mathematics and the teaching and learning of mathematics, and those beliefs sometimes aren't swayed by current evidence.
  5. How do we reflect on our learning? This is another problem we now know much more about, particularly due to Schoenfeld and his work on metacognition.
  6. How do we develop a mathematical attitude? This is still a challenge, and not just because some students say they don't like math. I think this problem might be closest to what Jo Boaler is currently trying to change with her focus on mindsets in learning mathematics.
  7. How do we coordinate students working together when the are at different levels of learning? Many teachers and scholars have worked quite hard on this problem and I feel like most teachers now see the benefit of heterogeneous ability groups. For more, I'd suggest Ilana Horn's book, Strength in Numbers.
  8. How do we create contexts for mathematizing? I think there's been a wealth of work in this area, from work based in Realistic Mathematics Education, work on word problems like that from Verschaffel, Greer, and de Corte, and, most recently, Dan Meyer's work. I could go on, as there are many more examples, and perhaps future work will give us a clearer picture about which contexts work best and why.
  9. Can we teach geometry by having the learner reflect on spatial intuitions? Maybe it's my lack of expertise in geometry education research, but I really don't know where we stand on this problem. Freudenthal seemed to be reaching in his article on this problem, and maybe a more tangible articulation of the problem would have helped me better judge any solutions we might have.
  10. How can technology increase mathematical understanding? Freudenthal admitted not being tech-savvy even in 1981 (he used "the ballpoint" as an example of technology that changed instruction, and not in an obviously historical way), but I think we now have numerous examples of tech that helps increase understanding. We also have a lot of examples of tech that doesn't, and I'm sure Freudenthal would have seen problems in our ability to judge the good from bad.
  11. How do we use a holistic approach to educational development for change? In his native Netherlands, Freudenthal would likely be pleased today to see his colleagues' commitment to design-based, participatory approaches to research. We have some of that here in the U.S., too, but we also struggle for a "scientific" approach to finding "what works" based on experimental studies. We also have too much faith in how standards affect change; if Freudenthal thought curriculum development for change was a wrong perspective, surely he'd think the same about standards. Those things are just part of a much bigger picture.
Looking at this list, I think we have a lot to be proud of. Even though Freudenthal's article wasn't some sort of directive or command to fellow and future math education researchers and teachers, many people over many years worked so we'd have some answers to these questions. Still, there's a gap between ''what the field of math ed knows'' and ''what a teacher does with this knowledge, if they know it," which hints at what might be a grand challenge of its own. I'd like to get to that, but in a later post. Next, I'll look at some of the grand challenges that I've seen others post on the web in response to NCTM's call for input.

Sunday, June 8, 2014

Schneider's From the Ivory Tower to the Schoolhouse, Chapter 5: Ideas Without a Foothold

In the first four chapters of From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education, Jack Schneider details how four ideas (Bloom's taxonomy, multiple intelligences, the project method, and Direct Instruction) traversed the gap between the education research world and K-12 classrooms. Now, in Chapter 5, he identifies counterparts to each idea that failed to make the leap: Krathwohl's taxonomy for the affective domain (the sequel to Bloom's Taxonomy for the cognitive domain), Sternberg's triarchic theory (which paralleled multiple intelligences), Wittrock's generative learning model (see Michael's post), and the behavior analysis model (similar to Direct Instruction in more ways than one). If my personal experience is any indication, Schneider has chosen these well, as I knew as a teacher about all the ideas in Schneider's first four chapters (to various degrees, anyway) but can't say I knew any of the four ideas compared in Chapter 5. To be honest, Chapter 5 served as my proper introduction to these latter ideas — not only did I not know of them as a teacher, I can't recall having learned about them in grad school, either.

In his review of Chapter 5, Michael Pershan takes the position that even though he hadn't heard of Wittrock's generative learning model, surely there existed some path by which he was at the tail end of some chain of Wittrock's influence. I think this is probably true; while teachers might only recognize Piaget and Vygotsky by name, the rise of the study of cognition and how we construct knowledge is the result of the work of many scholars, not just two. I think this falls under Schneider's concept of perceived importance: Piaget and Vygotsky seem important because so many scholars built upon their work, even if the scholars in that crowd remain nameless to us.

Still, it's difficult to say this is good enough. Even though it's not possible for a teacher (or anyone!) to have a direct connection to all available research, shorter paths would be preferable to long ones. I agree with Michael: teachers would likely benefit from knowing Wittrock and his work. But to what degree?

One of the things we learned from Schneider's first four chapters is that familiarity sometimes does not breed fidelity in education research. This felt most true in the multiple intelligences chapter, where some consultants seemed to play fast-and-loose with Gardner's theories, and I imagine the teachers who sat through those workshops or read those books played even faster-er and looser-er with multiple intelligences. Should we be worried that a little bit of knowledge is indeed a dangerous thing in education research?

I would be more worried if not for one thing: constructivist theories of learning tell us that not only to bits of knowledge matter, they're the stuff upon which more knowledge is constructed. In fact, there's a particular learning theory that addresses this called knowledge in pieces, and, if you can find it, it's worth reading Andy diDessa's 1988 chapter by that title. This should be of particular interest to Michael as the theory gives a nice way of explaining misconceptions, whether they be the ones we see in students or the ones we see teachers make when research finds its way to them by vague and indirect paths. In short, misconceptions aren't just the acquisition of "wrong" knowledge that needs to be confronted with "right" knowledge. Rather, knowledge in pieces says learners systematize their pieces of knowledge. What we think of as a "misconception" can be explained as a system of knowledge built upon pieces of available knowledge. The pieces aren't "wrong" and neither is the system, but as more pieces of knowledge get added we expect the system to adapt and become more sophisticated. Now, I admit that my understanding of the theory might be short a few pieces, but I think the key to wrapping your head around it is to force yourself to think knowledge exists with the learner, and nowhere else. Knowledge gets constructed from experience, not with the acquisition of knowledge from an external source. (See also: radical constructivism.)

Opening quote from diSessa's 1988 chapter

This leads us back to one of the ideas Michael mentioned in his post: teachers need exposure to research followed by opportunities to engage with the research more deeply. Teachers will take the pieces of knowledge they have — whether gained from teaching experiences, experiences engaging with research, or elsewhere — and systematize that knowledge in variously sophisticated ways. What we need, then, are opportunities for teachers to further systematize their knowledge. I'll talk about that in my next post, a review of Schneider's recommendations for improving research-to-practice.

Note: Michael Pershan (@mpershan) and I are reading Jack Schneider's book From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education. Our previous posts:


diSessa, A. A. (1988). Knowledge in pieces. In G. Forman & P. B. Pufall (Eds.), Constructivism in the computer age (pp. 49–70). Hillsdale, NJ: Lawrence Erlbaum Associates.

Sunday, June 1, 2014

Schneider's From the Ivory Tower to the Schoolhouse, Chapter 4: Direct Instruction

The fourth chapter of Jack Schneider's From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education represents a needed turn in the overall narrative of the book. As a bonus, this chapter will likely keep me from throwing around the phrase "direct instruction" in unintended ways.

Schneider's previous three chapters focused on Bloom's Taxonomy, multiple intelligences, and the project method. Each of those cases seemed to rely heavily on Schneider's constructs of philosophical compatibility and transportability. In other words, fidelity of implementation didn't seem to matter much: teachers adoption of the research seemed tied to their freedom to interpret and implement the research in whatever way they saw fit. In more than a few instances, Schneider leaves the reader to question if the research has been implemented with any fidelity at all, or if teachers are adopting it in name only.
In this chapter, titled Lessons of Last Resort, Schneider tells the story of Direct Instruction. I've heard and used the term direct instruction (little "d" little "i") to simply describe teaching as telling, but it has a more specific research heritage exending back 50+ years. The researcher there from the beginning is Siegfried Engelmann, seen here:

Unlike Bloom's Taxonomy, multiple intelligences, and the project method, Englemann's Direct Instruction works (with the research to show it) when teachers are philosophically compatible with the method and they implement it with fidelity. The actual effectiveness of research wasn't addressed in Schneider's first three chapters, but it is here because it's one of the big reasons for Direct Instruction's success.

This success isn't something that makes some progressive educators very comfortable, as they resist the scripted nature of the curriculum. These progressive educators are usually in schools where illiteracy and innumeracy isn't a persistent problem, and they're given autonomy to choose other, more philosophically compatible curriculum and methods. (To be clear, just because Direct Instruction has been shown to be effective, that doesn't mean it's the only effective thing, or the most effective. Also, it should go without saying, showing something to be "effective" is a tricky business, even when we agree what "effective" means.) But in schools where illiteracy and innumeracy persists, often in low-income schools with underrepresented populations and difficulties finding skilled teachers, Direct Instruction is more popular. Schneider addresses the issue of philosophical compatibility:
In addition to its effect on teacher authority, scripting also promised to reduce the responsibilities of those in classrooms. Working with a program like Direct Instruction, teachers would no longer be responsible for lesson design, for expertise about children, or for the task of dealing with the uncertainty of classroom life. As Direct Instruction promoters put it on their Web site: "The popular valuing of teacher creativity and autonomy as high priorities must give way to a willingness to follow certain carefully prescribed instructional practices." And as Englemann put it: "The teacher is a teacher—not a genius, an instructional designer, or a counselor. The teacher must be viewed as a consumer of instructional material." Engelmann saw this aspect of Direct Instruction as occupationally realistic, and he may have been right. But reducing teacher responsibility also raised serious philosophical compatibility issues insofar as it threatened teacher professionalism. (p. 122)
You might be reading this right now and saying to yourself, "No way. I'd never use this stuff." That's the philosophical incompatibility talking. There's reasearch for that, too: reform curricula might be good, but the results aren't nearly as good when placed in the hands of a traditional teacher. I believe vice-versa has been found to be better, but still not as good as reform curriculua with reform teachers. But where do we draw the line between philosophical compatibility and the need for teachers to be open minded? To be learners? As professionals, when should our philosophies give way to what we can gain from research, regardless of compatibility?

I don't have an answer for this question, but perhaps Michael Pershan (@mpershan) will have some thoughts in his reply. If you haven't been following along, we've been reading the book together and here are our posts so far:

Schneider's From the Ivory Tower to the Schoolhouse, Chapter 3: The Project Method

Michael Pershan and I, in our chapter-by-chapter review of Jack Schneider's From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education, now take issue with the project method and, more generally, the promotion of academic research. I'll make a comment or two about the chapter and the project method, and then turn to responding to Michael.

Of the chapters and ideas Schneider has presented thus far (Bloom's Taxonomy, multiple intelligences), it's hardest to get a grip on the project method. The reason comes in the second paragraph: "The project method is so well accepted that modern educators simply view it as property of the educational commons" (p. 79). I found myself grasping to Schneider's few hints about what education was like before the project method, and imagined classroom activity based around lecture, exercises, drills, and recitations. Certainly there were some counterexamples, but I'll take Schneider's word that William Heard Kilpatrick made the project method famous, and made himself famous in the process.

In the last chapter about multiple intelligences, Schneider discusses Howard Gardner's efforts to promote his work: writing books published by popular presses, making speaking engagements, and supporting the work of those using (and sometimes misusing) the idea of multiple intelligences. In this chapter, a considerable amount of attention is given to Kilpatrick's desire to achive "power and influence" (Kilpatrick's diary, as cited by Schneider, p. 81). In his introduction, Schneider gave us four characteristics of research that traverses the divide between research and practice: perceived significance, philosophical compatibility, occupational realism, and transportablility. Here we seem to be concerned with a characteristic not of the research, but of the researcher. I don't feel like Schneider makes the distinction entirely clear, but I think you can relate the ego and ambition of the researcher to the perceived significance of the research.

In his post, Michael takes issue with Kilpatrick's quest for educational fame. Michael used "It's the Celebrities That We Need to Doubt" as the title of his post and warns us, "Famous people become famous because they want to be famous, and we need to judge their ideas with the skepticism that sort of person deserves." Fame can be a tricky thing in academia. In an enlightening (yet private1) Google+ conversation last year, I heard from several faculty members that despite the stated requirements for publishing, teaching, and service, what your department and university would really love is for you to help make them famous.

Note the difference between making your university famous and making yourself famous. Teachers College didn't need much help from Kilpatrick to make it famous, and Schneider makes it clear that Kilpatrick was interested in his own fame, hoping to be given the same esteem and recognition that Dewey had achieved. I share Michael's skepticism of self-promoters. In my teaching career here in Colorado, the only researcher I heard much about was Robert Marzano. Marzano runs an independent research lab here in Colorado and does work throughout the country. He sells lots of books, workshops, and "customized educational services." In grad school, on the other hand, I hear next to nothing about Marzano's work. I have a sense that Marzano has done good work, but perhaps quality has wavered as he's grown his operations. I have an even stronger sense, however, that Marzano's work just doesn't interest academia because it's not from academia, and he's not in academia. Marzano made himself a product and that's not a welcomed move by (at least some) people in scholarly circles.

I can think of a few other makes-some-people-uneasy examples even closer to academia. One is the Institute for Learning at the University of Pittsburgh. Founded by Lauren Resnick, IFL offers workshops, contracts with districts for professional development, and self-publishes its research. One key product for them is Accountable Talk®, and yes, I have to put that registered trademark symbol there because they trademarked it. I think some might look at Jo Boaler's effort with some skepticism, and Dan Meyer attracts some doubters, too. (It sure sounds like Kilpatrick would have loved being recognized for a well-watched TED Talk.) This might make readers of this blog uncomfortable, but I wouldn't doubt there are teachers who are skeptical of teachers using social media, thinking we're just in this for the fame.

Some of this sentiment is rooted in a culture spanning K-12 and higher education that says we educators are supposed to be humble, to be selfless, and to be dedicated to the service of others. I admit to feeling this way: just let me serve the public and, in return, let me be supported by the public. In my current work with teachers, I'm happy the National Science Foundation provides the funds for us to work together, rather than doing the work for the district on a contract basis. I don't want the role of salesman. That said, there's some unclear middle ground between this culture and edupreneuership. For example, I've seen some negative reactions on Twitter towards those who try to sell things on Teachers Pay Teachers, yet positive reactions towards those who have self-published a book on Amazon or co-authored something for NCTM.

Yet somewhere between the selfless and self-promoting cultures there needs to be the realization that if we're interested in research being taken up by K-12 educators, it simply isn't enough to let the science speak for itself. If it makes people feel better, think of it as "outreach" instead of "marketing," and "sharing" instead of "promotion." Schneider gives considerable credit to Gardner and Kilpatrick's efforts to widely share/promote their work for the success of multiple intelligences and the project method. Now that sharing is easier than ever, I'm hopeful that we'll see more blending of the research world and the practice world, and what might have been seen as self-promotion in Kilpatrick's day morphs into a genuine practice of a sharing-based educational community.

Note: Michael Pershan (@mpershan) and I are reading Jack Schneider's book From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education. Our previous posts:
Chapter 1: Bloom's Taxonomy (Michael's post, my reply) Chapter 2: Multiple Intelligences (My post, Michael's reply)

  1. I love that Google+ offers so much flexibility to make conversations public vs. private, but I'm frustrated by the number of high-quality posts shared only in small circles of math educators. But that's another post for another day. 

Wednesday, May 28, 2014

Schneider's From the Ivory Tower to the Schoolhouse, Chapter 2: Multiple Intelligences

In my last post I reviewed the first chapter of Jack Schneider's From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education. There Schneider describes how Bloom's Taxonomy, despite never being designed for K-12 use, managed to make it way into practice at all educational levels. In Chapter 2, Schneider looks at Howard Gardner's Multiple intelligences, another idea that made its way from the research world into K-12 classrooms.

Howard Gardner (CC BY-NC-ND by The Aspen Institute)
Schneider suggests that the success of multiple intelligences is as much a product of an age as it is a product of a scholar. It certainly helped that Gardner was a Harvard professor with a MacArthur "genius grant," but the year of publication — 1983 — plays a big part in the theory's success. The U.S. was well into the "back to basics" era that began in the 1970s, and 1983 was the year A Nation at Risk declared that the country's future depended on the reform of schools to make them economic drivers, namely by focusing on math, science, and technology. This focus on curriculum, especially a focus on a narrow part of the curriculum, is not what progressive educators want to hear. Progressives, in the tradition of Dewey, want to focus on students and their experiences, where curriculum plays the lesser role as merely the means to potentially many learning ends.

According to Schneider, many of these progressive educators were in independent/private schools and they became the early adopters of Gardner's theory. For those schools and teachers, multiple intelligences was a way to distinguish their philosophical stance on education from that of the oncoming (and still ongoing) accountability movement in public schools. Tuition-paying parents certainly didn't object to the idea that their children could be intelligent in more than one way, and the theory seemed to validate the idea of getting a well-rounded education.

The growth of multiple intelligences beyond independent schools, says Schneider, is owed to the transportablility of the theory. Like Bloom's Taxonomy, multiple intelligences is summarized by a limited set of categories with seemingly self-evident descriptions. It's not specific to a particular content area or grade level and matches what teachers see in practice: that different students have different ways of learning and develop different kinds of talents. As with Bloom's, transportability comes with risks of misinterpretation, as Schneider describes:
Multiple intelligences was a theory with different uses for educators. It could challenge the validity of tests, open up standardized curricula, and defend cherished beliefs about teacher professionalism and student ability. But whatever the use, it was a theory philosophically compatible among public school teachers. And it was highly transportable—seemingly easy to understand from the names of its "intelligences" alone. The ironic downside of this, of course, was that it could also be used as a bulwark against real deliberation or debate. As Gardner himself wrote, "It is possible to wave the MI flag without having to think, change, or grow." Though likely not a majority, that was certainly true for some. (p. 64)
In the latter part of this chapter I was surprised and impressed by Schneider's description of how consultants and professional developers played a major role in spreading the use of multiple intelligences. Schneider gets my skepticism of all-too-typical PD:
Competing with one another for often lucrative contracts, third-party providers have a strong incentive to entertain their clientele without asking too much in return, and to develop a message general enough that it can be adapted in multiple settings. Thus, despite research indicating that effective professional development is time-intensive, context-specific, and content-rich, a great deal of training relies on traditional methods of delivery and is strongly shaped by consumer desire. (p. 69)
For those looking for an educational disaster narrative, Schenider's description of how some consultants and authors twisted multiple intelligences is interesting reading. Gardner's role in this is equally interesting, as he seems to sway between defending his theory and supporting those who sometimes misinterpret it for their personal gain. Schneider asserts that Gardner "was not in control [of the interpretation of multiple intelligences], and that perhaps he never had been" (p. 73).

If there was one thing I wish Schneider would have expanded upon, it would have been the commingling of multiple intelligences and learning styles. It's addressed in a couple of paragraphs but only briefly. I'd guess that Schneider could have a chapter dedicated to how theories of learning styles became pervasive in K-12 education, but it might have been too similar and perhaps redundant next to a chapter on multiple intelligences. Similarly, other often-believed theories (like right-brained/left-brained) might have fit in Schneider's book, but I'll take this chapter as representative of the lot. I can always look for additional commentary elsewhere, such as in a recent blog post titled Can Teachers Stop Believing in Nonsense? that addresses common K-12 misapplications of neuroscience.

As I spend more time in the research world, I have an opportunity to not only learn more about theory, but to get to know the researchers behind those theories. Some are at peace with the idea that others will "do what they will" with their work, while others want more control. Gardner took an active role in promoting his work, either directly or through the work of others, and I think he was right to do so. Academics are notoriously poor marketers of their work, which causes useful and legitimate research to get lost among the better-promoted work of think tanks or others whose marketing exceeds their scholarship. Part of the problem is a mismatch of incentives, but I think Gardner, for his struggles, did get some things right: write for a wide audience, advocate for your work, use your work to advocate, and support others who make productive adaptations to your work.

Reminder: Michael Pershan (@mpershan) and I are reading this book together, and for this chapter it's his turn to reply to my post. Keep an eye on his blog at for his follow-up.