How the Race (to the Top) Was Won (Part 1 of 2)

Race to the Top (RTT), the foremost education policy instrument under the Obama administration, was introduced in 2009 as part of the American Recovery and Reinvestment Act. The first two winning states, Delaware and Tennessee, were announced in the spring of 2010. At that time I wrote a paper about RTT for a policy class but didn't blog about it. Since RTT has continued with more rounds of state competition and a new RTT program for school districts, I hope there's still some relevance in sharing some of what I learned about RTT and what states did to score well on their applications.

Using policy definitions developed by McDonnell and Elmore (1987), RTT is a near-perfect example of an inducement, where money is exchanged for action. Inducements are a simple model:

RTT adds a layer of policymaking to this simple process. In the model below, the state departments of education have all policy instruments at their disposal. The inducement exists between the federal and state policymaking bodies, and not necessarily between the state and the local education authority (LEA). (The new RTT for districts will obviously change this arrangement.) Regardless of the instrument(s) used by the states, the goal, as defined by the U.S. DoE, is for states to "[lead] the way with ambitious yet achievable plans for implementing coherent, compelling, and comprehensive education reform" (U.S. Department of Education, 2010d). Additionally, the DoE clearly states that "Creativity and innovation are rewarded in this competition" (U.S. Department of Education, 2010c. p. 15).

In the ideal RTT scenario, the competition would look like this:

1. The U.S. DoE rewards the states with the most promising reforms.
2. Winning states would enact and enforce new education policies.
3. The effects of the new policies would be measured.
4. Policies that prove to be successful would be replicated by other states.

There are two significant problems with emphasizing creativity and innovation in RTT. First, creativity and innovation necessitates deviation from proven reforms. You can't be creative by saying, "We're going to do what we know works." Delaware and Tennessee's "innovative" reforms (a label worth questioning) may have helped them win Phase 1 of RTT, but it may some time before we know if the reforms perform as intended. Implicit in the RTT competition is an assumption that established, effective reforms are too few, too expensive, or too difficult to scale, so RTT challenges states to create new reforms that might be cheaper or more easily implemented. This creates a condition where RTT money gets awarded for the potential of a policy, and not its past, proven effectiveness.

The second problem with emphasizing creativity and innovation is that RTT was not structured as an brainstorming, anything goes kind of policymaking process. RTT comes with a detailed scoring rubric, and any state wishing to score well obviously wrote policies to satisfy the rubric. How does that encourage creativity or innovation? In addition, states or districts applying beyond the first round will likely replicate the highest-scoring applications from Phase 1. This means that policies developed as part of RTT Phase 2 and later are likely to be less diverse, less creative, and less innovative, but no more proven.

So how did states manage this balance of creativity versus scoring high on the rubric? Thankfully, the rubric, the applications, and the judges' scorecards are all publicly available, so we can see exactly what each state proposed in their application. The analysis in this post will answer the following questions:

1. Where in the RTT rubric could states score the most points?
2. For the portions of the RTT rubric identified in #1, which states scored highest?
3. For the states identified in #2, what did their application propose and what were the judges' comments?

Race to the Top Scoring

Here is a summary of RTT Phase 1 scoring:

Selection Criteria	Points Possible	Percent of Total	Average Score (Points)	Average Score (Percent)	Standard Deviation (Points)
A. State Success Factors	125	25	90	72	18.03
B. Standards and Assessments	70	14	61	88	9.63
C. Data Systems to Support Instruction	47	9	33	70	7.04
C. Data Systems to Support Instruction	47	9	33	70	7.04
D. Great Teachers and Leaders	138	28	91	66	21.5
E. Turning Around the Lowest-Achieving Schools	50	10	36	72	11.19
F. General	55	11	37	68	12.96
Competitive Preference Priority 2: Emphasis on STEM	15	3	11	73	6.73
TOTAL	500	100	359	72	67.22

(Source: U.S. Department of Education, 2010a, 2010b)

Looking at how the points on the rubric are allocated, it's clear that for any state to do well they'd need to score well on Section D, "Great Teachers and Leaders." It was worth 28 percent of the 500 total points, and we now can see that of all the sections, the fewest points (as a percent) were awarded in this section. That means there was a lot of potential upside here, and we can dig into the details of "Great Teachers and Leaders" to see which states scored best. In the table below you'll find all the subsections of Section D, along with the scores of the eight states (DE, TN, GA, SC, RI, KY, LA, and KS) who had the top score (in bold) in at least one of those subsections.

Selection Criteria	DE	TN	GA	SC	RI	KY	LA	KS
Overall RttT Phase 1 Rank	1	2	3	6	8	9	11	29
(D) "Great Teachers and Leaders" Score	86%	83%	81%	82%	88%	80%	89%	63%
(1) Providing high-quality pathways for aspiring teachers and principals	82%	71%	70%	74%	84%	94%	89%	32%
(2) Improving teacher and principal effectiveness based on performance	87%	91%	86%	91%	94%	76%	90%	56%
(2)(i) Measuring student growth	88%	100%	48%	88%	80%	84%	96%	56%
(2)(ii) Developing evaluation systems	85%	91%	81%	89%	96%	83%	91%	68%
(2)(iii) Conducting annual evaluations	92%	100%	96%	90%	100%	58%	92%	68%
(2)(iv) Using evaluations to inform key decisions	86%	87%	91%	93%	93%	76%	89%	45%
(3) Ensuring equitable distribution of effective teachers and principals	85%	74%	87%	73%	79%	73%	90%	89%
(3)(i) Ensuring equitable distribution in high-poverty or high-minority schools	83%	68%	93%	68%	92%	72%	88%	85%
(3)(ii) Ensuring equitable distribution in hard-to-staff subjects and specialty areas	88%	82%	78%	80%	60%	74%	92%	94%
(4) Improving the effectiveness of teacher and principal preparation programs	81%	90%	73%	81%	90%	77%	86%	66%
(5) Providing effective suport to teachers and principals	95%	75%	75%	79%	84%	92%	84%	80%

(Source: U.S. Department of Education, 2010a)

Just by reading the titles of the subsections above, you can recognize some of the most contentious areas of education policy we've seen the past several years. So not only is this part of the RTT application about high points, it's high-stakes. If the greatest opportunity for improvement in education truly lies in this area, it will be critical to get these policies right. In Part 2 of this post, we'll look at each subsection and the application from the state with the highest score in that area. Some of the policies are sound, but some aren't, and sometimes the judges' comments indicate divergent interpretations of both the application and the rubric.

References

McDonnell, L. M., & Elmore, R. F. (1987). Getting the job done: Alternative policy instruments. Educational Evaluation and Policy Analysis, 9(2), 133-152. Retrieved from http://www.jstor.org/stable/1163726

U.S. Department of Education. (2010a). Detail chart of the Phase 1 scores for each State. Retrieved from http://www2.ed.gov/programs/racetothetop/phase1-applications/phase1-scores-detail.xls

U.S. Department of Education. (2010b). Race to the Top Scoring Rubric Corrected. Washington, D.C.: U.S. Department of Education. Retrieved from http://www2.ed.gov/programs/racetothetop/scoringrubric.pdf

U.S. Department of Education. (2010c, May 27). Race to the Top Program: Guidance and Frequently Asked Questions. Retrieved from http://www2.ed.gov/programs/racetothetop/faq.pdf

U.S. Department of Education. (2010d, April 16). Race to the Top Fund. ED.gov. Retrieved June 6, 2012, from http://www2.ed.gov/programs/racetothetop/index.html

Open Access Publishing in Mathematics Education

As I write this, the White House petition to require free access over the internet to scientific journal articles arising from taxpayer-funded research is within 150 signatures of the 25,000 needed to guarantee a response from the White House. If you're unfamiliar with the petition, this video concisely explains the issue:



Most of the advances in open access publishing seem to be in the natural and medical sciences -- mathematics, biology, medicine, etc. Much is this is due to a policy by the National Institutes of Health (NIH) that requires publications from NIH-funded research be made available to the public, and the hope of the petition is that a similar policy would spread to other government funding agencies. Given that a significant amount of education research is funded by the National Science Foundation (NSF), education researchers are going to have to think about their open access publishing options should policies require open access to publicly-funded research.

I'm a teacher, not a researcher. Should I care about open access to research?

Yes! Few things annoy me more than the assumption that teachers should not read or take any interest in published education research. I strongly believe that the more researchers think about teachers as part of the audience for their research, the more relevant that research is likely to be and the more quickly we can implement the results. So if you're a teacher and you come across a research article, pay attention. If it makes no sense to you or seems totally irrelevant to you as a teacher, there's probably something wrong. You would be doing a great service to bring that article and your problems with it to the attention of the research community, and researchers should welcome your input. Right now a lot of high-quality research hides from you in for-profit, closed journals, which I believe has allowed the goals of teachers and researchers to drift apart. With open access journals and greater communication via social media, I hope the divide between teachers and researchers can come together with greater frequency.

Current Top-Tier Mathematics Education Research Journals

In mathematics education, the following four journals are often seen as the most prestigious. Let's look at their current publishing policies:

Journal of Research in Mathematics Education - JRME is NCTM's research journal and  probably the top journal in the field. Unfortunately, they have a very author-unfriendly publishing policy:

Assignment of copyright for the article to the National Council of Teachers of Mathematics is required as a condition of publication. After acceptance by JRME, a manuscript may not be published elsewhere, including on the internet, without written permission from NCTM. Each author of a paper published in JRME will receive five complimentary copies of the issue in which the paper appears.

Wow, five complimentary copies? With those I can freely distribute my work to 5 people, or approximately 0.0000002% of worldwide internet users, all of whom can read this measly blog page.

Educational Studies in Mathematics - This journal was founded by Hans Freudenthal in 1968 and is currently published by Springer. Although Springer is an enormous publishing company with a vested interest in a traditional publishing model, they are making efforts to find ways to increase access while still making a profit. Authors have a choice: (a) Transfer their copyright to Springer or (b) opt into Springer's "Open Choice" program, which makes articles freely available on SpringerLink and allows the author to retain copyright and publish under a Creative Commons Attribution License. The catch? Springer charges the author a $3000 fee.

International Journal of Science and Mathematics Education - This journal is also published by Springer and has the same "Open Choice" option as ESM.

Mathematical Thinking and Learning - This journal is published by Routledge, part of the Taylor & Francis Group. Their copyright agreement (PDF) includes the classic language about why authors should transfer copyright (and when I say "classic," I mean old, as in pre-internet): "The transfer of copyright from author to publisher must be clearly stated in writing to enable the publisher to assure maximum dissemination of the author's work." The thought that putting written work in an expensive journal distributed to a relatively small number of people and institutions "assures" a wider distribution than the open internet is plainly laughable. The copyright agreement does throw a few bones the author's way with these three exceptions:

1. Authors can copy their own article for their use in classrooms.
2. Authors can reuse the work in a textbook they might author.
3. Authors can copy their work for internal distribution within their institution.

Exception #2 is not to be overlooked -- some publishers will not grant that exception. I know one researcher who wanted to re-use an article he'd written as a dissertation chapter and was denied, forcing him to start the research and writing anew.

Current Open Access Journals

Assuming the journals above don't convert themselves to an open access publishing model (one without $3000 fees), the most immediate option for publishing under an open access mandate would be in an journal that's already open access. No, these journals don't have the history or prestige that the above journals have, but I do get the sense that tools like Google Scholar are making the journal name less relevant than in the past. Many of these journals have emerged in just the past 5-10 years, and I'll limit the list below to those that publish primarily in English and appear to receive submissions from U.S.-based researchers. All the journals found below were indexed in the Directory of Open Access Journals (DOAJ).

I divide open access journals into two main camps -- those where the publisher takes copyright, and those where the author retains copyright. Given the open nature of these journals, I imagine some negotiation about copyright would be very acceptable, particularly if using something like the SPARC Author Addendum.

Author Copyright

International Journal for Mathematics Teaching and Learning - IJMTL is a joint publishing effort between Plymouth University, UK, and the College of Nyiregyháza, Hungary. Their author guidelines say nothing about copyright, but authors are expected to do their own copy editing and formatting of their final article. The articles I looked at made no mention of copyright or licensing, so I assume authors have retained copyright and have the right to assign a Creative Commons license if they wish.

Journal of Statistics Education - JSE has been published by the American Statistical Association since 1993 and clearly indicates the author's copyright on each article.

Journal of Urban Mathematics Education - JUME is edited by David Stinson at Georgia State and appears to be one of the higher-quality open access efforts, publishing articles by William Tate, Rico Gutstein, Megan Staples, Jere Confrey, Michael Battista, Jo Boaler, and others. Authors retain copyright with first publication rights granted to JUME.

Numeracy - This journal specializes in quantitative literacy and is hosted by the University of South Florida. Authors retain copyright under a Creative Commons Attribution 3.0 license. There are no publication charges.

Philosophy of Mathematics Education Journal - Edited by Paul Ernest at the University of Exeter, UK, this journal has existed since 1990 and the copyright notice reads: "All materials published herein remain copyright of the named author(s), or of the editor if unattributed. Permission is given to freely copy the journal contents on a not-for-profit basis, provided full credit is given to the author and the journal." This sounds much like a "legal lite" interpretation of a Creative Commons Attribution - Non Commercial license, although there's no explicit mention of derivative works.

Pythagoras - This is the journal of the Association for Mathematics Education of South Africa, in existence since 1980. The copyright notice on a recent article indicates that the author retains the copyright and the work is licensed under a Creative Commons Attribution license.

Technology Innovations in Statistics Education - TISE is edited by Robert Gould at UCLA and authors retain copyright with publication under a Creative Commons Attribution - Non Commercial - Share Alike license.

The Teaching of Mathematics - This is published by the Mathematical Society of Serbia and makes no mention of copyright on their site or on published articles, so I assume copyright would stay with the author.

Publisher Copyright

Contemporary Issues in Technology and Teacher Education - Part of this journal includes articles about technology and mathematics education, with editorial review organized by AMTE. The publisher retains copyright to published articles.

International Electronic Journal of Mathematics Education - IEJME is published by Gökkuşağı, a Turkish publisher, and dates back to 2006. Their author guidelines don't say anything about who retains copyright of the published articles, but the journal itself indicates the copyright is held by Gökkuşağı.

Journal of Mathematics Education at Teachers College - This is a good looking journal with submissions from some well-known authors. The submission guideline page says nothing about copyright, but the journal itself and each article claims copyright for the publisher.

Journal of STEM Education - This journal requires the author pay a fee ($395 for the first 8 pages and author's bio, then $35 for each additional page) and also requires the transfer of copyright.

Statistics Education Research Journal - This international journal appears to include a wide variety of content but requires authors to transfer copyright.

The Mathematics Educator - TME is a student-produced journal from the University of Georgia and was first published in 1990. Despite the maturity of this journal, copyright is very unclear -- the site says nothing about transferring copyright, and the journal itself claims copyright for the publisher in the front matter, but nothing on articles themselves, and articles are available individually.

The Mathematics Enthusiast - Formerly known as the Montana Mathematics Enthusiast, this journal says nothing on the site about transferring copyright, but the articles themselves indicate a copyright held by the publisher.

Conclusion

It's difficult for me to predict exactly how an open access mandate would affect current journals. Those top four journals, because of their prestige, might not change a thing and hope to get submissions from authors who aren't funded by major federal agencies. Some of the open access journals are obscure now and will likely stay that way, at least to U.S. researchers. What I'd like to see is some of the currently closed "second-tier" journals open themselves. Some, like The Journal of Mathematical Behavior (an Elsevier publication), isn't a likely candidate. For a journal like For the Learning of Mathematics, opening access might be easier. (FLM already has a FAQ including the question, "Can I reprint an FLM article on my web site / anthology / lunch box?" with the simple answer, "If you are interested in reprinting articles that appear in FLM, please contact the managing editor.") Some journals already have policies that would seem to pull them in the direction of open access. Teaching Statistics, for example, is closed but allows authors to retain copyright so long as they give an exclusive license to publish to the journal. I don't know what good it is to have a copyright but no right to publish, but some tweaking of those policies might turn such a journal into something open.

Then again, maybe math education researchers will gravitate towards current large open access repositories. The article Number Concepts without Number Lines in an Indigenous Group of Papua New Guinea caught my eye not just for its content, but the fact it is published in PLoS ONE, a journal that's flourished publishing open access science and medicine content, not necessarily education-related articles. But there's no reason PLoS ONE can't expand its scope, something it's likely to do if new governement open access policies demand more open publications in more content areas.

RYSK: Gutiérrez's (Re)Defining Equity: The Importance of a Critical Perspective (2007)

This is the eighth in a series of posts describing "Research You Should Know" (RYSK).

Do you ever find yourself talking about something, defending something, or promoting something when you suddenly realize you don't have a good definition of that thing?

In one way or another, I've been thinking about equity in math education ever since I was an undergraduate. I remember debating the value of "equality of opportunity" versus "equality of outcomes," and getting a sense for how the NCTM Standards prescribed a type of school mathematics for all students. Here at CU-Boulder, issues of equity and social justice are never far away. But what, exactly, do we mean by equity in math education? And why is it important?

Rochelle Gutiérrez focuses on issues of equity as an associate professor of mathematics education at the University of Illinois at Urbana-Champaign. Even though she's been publishing on issues of equity for well over a decade, in 2007 she wrote a book chapter titled, (Re)Defining Equity: The Importance of a Critical Perspective. In that chapter, she argues why we need a definition of equity that gives teachers and researchers a clear sense of purpose.

When equity is loosely defined, it comes under attack from several directions. First is a belief that not all students can learn, and that mathematical proficiency has more to do with natural ability than with effort. The second threat to equity is a "deficit theory" towards groups of students that haven't had much historical success in mathematics, whether that deficit is seen as biological or cultural. The third threat to equity, says Gutiérrez, comes from within the research community itself: so many issues get covered under the umbrella of "equity" that few of them get the kind of focused attention they need, even while many agree equity is important. As Gutiérrez puts it:

Perhaps the lack of a clear definition is what contributes to a general consensus that equity is worth striving for, everyone having his or her own vision of what it means. However, having a poorly defined target means we are only sure we are moving toward it when, in fact, we are very far away. (p. 38)

Gutiérrez argues that we should leave behind the traditional "excellence versus equity" and "traditional versus reform" debates in favor of a new perspective: dominant versus critical. Instead of teaching mathematics that "reflects the status quo in society, that gets valued in high-stakes testing and credentialing, that privileges a static formalism in mathematics," (p. 39), we should be favoring critical mathematics, that which "squarely acknowledges the positioning of students as members of a society rife with issues of power and domination" (p. 40). This includes using math to examine social and political issues, to highlight perspectives of different cultures, and to challenge the view that mathematics is a static entity. Gutiérrez does not wish to create a dichotomy here -- in fact, she argues the importance of learning dominant mathematics because it can help students better understand and criticize the world.

With that perspective in mind, Gutiérrez defines equity. First, she warns not to confuse it with equality; whereas equity implies "justice" or "fairness," equality implies "sameness." Gutiérrez is *not* arguing that all students should experience the same instruction using the same materials, or that we should expect all students to have the same outcomes. Gutiérrez fully recognizes that, within any group, experiences and outcomes will vary, and that some students will have interests that lead them away from mathematics. That's okay. Instead, she says equity in mathematics should consist of three main parts:

1. "Being unable to predict students' mathematics achievement and participation based solely upon characteristics such as race, class, ethnicity, gender, beliefs, and proficiency in the dominant language" (p. 41, emphasis original). To clarify, Gutiérrez says, "I contend that only when there is sufficient variation within groups and no clear patterns associated with power or status in society between groups can we conclude that this aspect of equity is being addressed" (p. 42, emphasis original). As for measuring achievement, Gutiérrez says we should use standardized tests (but not exclusively), because those are often the tools we use to grant power to individuals.
2. "Being unable to predict students' ability to analyze, reason about, and especially critique knowledge and events in the world as a result of mathematical practice, based solely upon characteristics such as race, class, ethnicity, gender, beliefs, and proficiency in the dominant language" (p. 45, emphasis original). It's this aspect of equity that Gutiérrez uses to stress the critical aspects described above.
3. "An erasure of inequities between people, mathematics, and the globe" (p. 48, emphasis original). Gutiérrez claims "This aspect of equity addresses the fact that having equal access to cultural capital and critical stances to society are necessary but insufficient conditions for change" (p. 48). While this aspect of equity is by far the most difficult to measure, and may not happen in our lifetimes, it should be the key goal of any long-term reform in mathematics education.

Gutiérrez closes her paper with this paragraph:

It might be the case that the first two aspects of equity must be addressed before we would see any changes in the third aspect. That is, students who gain both (1) dominant and (2) critical mathematics identities will lead to different kinds of mathematicians in the academy, thereby changing what counts as mathematics as well as how it is evaluated. The important thing to consider in this (admittedly simplistic) model is that neither the first nor the second aspects of equity are sufficient to redress injustices in the world. Students need to be able to do both -- be able to play the game of mathematics that is currently associated with power and intellectual potential, and be able to change the game of mathematics to serve a better society. (p. 49)

References



Gutiérrez, R. (2007). (Re)defining equity: The importance of a critical perspective. In N. S. Nasir & P. Cobb (Eds.), Improving access to mathematics: Diversity and equity in the classroom (pp. 37-50). New York, NY: Teachers College Press.

RYSK: Staples's Supporting Whole-Class Collaborative Inquiry in a Secondary Mathematics Classroom (2007)

This is the seventh in a series of posts describing "Research You Should Know" (RYSK).

In her book What's Math Got to Do With It?, Jo Boaler recounted her first contact with the math wars. A group of parents at a local school had organized against the adoption of reform textbooks, and were telling students that if they took the classes with the new books, they wouldn't be eligible for college. Apparently the parents had called admissions offices and asked a question like, "Would you accept a student who had not taken any math in high school but had just talked about math?" (Boaler, 2008, p. 33). Of course the colleges said no, and from that question and response parents based a claim about reform texts and college admission.

If we try to put angry politics aside for a moment, where did the parents in Boaler's story get the idea that reform math was all about talking about math? Certainly that thought was not a total fabrication. In fact, the learning theories that influenced much of the reform math movement led teachers and researchers to think about how to structure classrooms in ways that maximized learning, and that led to an attention to classroom discourse -- the teacher-student and student-student speaking and writing that happens in classrooms. By studying classroom discourse, we can gain key insights about what and how math is learned, and how our experiences and our environment affect that learning. (This, I believe, is not unique to reform classrooms -- all good math teachers and students pay careful attention to how we talk about and otherwise communicate mathematics.)

Megan Staples, a former advisee of Jo Boaler, is an assistant professor at the University of Connecticut specializing in mathematics education and classroom discourse. In her article Supporting Whole-Class Collaborative Inquiry in a Secondary Mathematics Classroom (2007), Staples takes on the challenges of being the teacher and supporting students in "collaborative inquiry." Instead of simply viewing collaboration as working in groups, or cooperating to complete a task, Staples says collaboration, "implies a joint production of ideas, where students offer their thoughts, attend and respond to each other's ideas, and generate shared meaning or understanding through their joint efforts" (p. 162). As for inquiry, Staples sees it as a way of "engaging with and making sense of the world" (p. 163), and applies the term to "both inquiry into mathematics and inquiry with mathematics" (p. 163). Finally, and perhaps most importantly, is how Staples defines learning mathematics itself. Using a situative perspective, Staples sees mathematics as a cultural practice, where "learning results from, and is evidenced by, student participation in both standard and disciplinary practices (e.g., justifying, representing algebraically) and an array of other practices of mathematical communities (e.g., questioning, communicating, informal reasoning)" (p. 163). With these conceptions of collaboration, inquiry, and learning mathematics in mind, Staples conducted a year-long observation and analysis of Ms. Nelson, a veteran, award-winning teacher known for her dedication to reform mathematics principles. The class Staples analyzed was called "Math A," a lower-track 9th grade class for students with a history of low performance in traditional mathematics classrooms.

Staples's analysis yielded two models for teaching that support student participation in class discussion. The first model describes the role of the teacher in a whole-class discussion, while the second describes how Ms. Nelson increases the class's ability to collaborate over time. I'll present Staples's findings in outline form, with attention to specific recommendations for teachers wishing to support collaborative inquiry in their own classrooms. (Be patient -- the original article is 57 pages long, after all.)

1. Model 1: The teacher's role in supporting whole-class collaborative inquiry
   
   1. Supporting students in making contributions
      
      • Eliciting student ideas -- Instead of just asking questions like, "Why?" or "How do you know?," Ms. Nelson presses students to share with comments like, "Come on, I'm really interested, come on, you can do it" (p. 175), gave students adequate time to formulate explanations, and offered participation points as a reward for contributing ideas.
      • Scaffolding the production of student ideas -- Ms. Nelson helps direct struggling students to use multiple representations and models, such as number lines, graphs, diagrams, etc. The key, says Staples, is to provide structure for the mathematics without constraining how the students will work out the mathematics (p. 178).
      • Creating contributions - Ms. Nelson treated incomplete and incorrect contributions by students the same as correct ideas, saying things like, "Remember the idea is to go up and give us some good discussion...that helps the class move along regardless of whether it's right or wrong, it enables us to have good discussion" (pp. 178-179).
   2. Establishing and monitoring a common ground
      
      • Creating a shared context -- Ms. Nelson focused on creating shared contexts among students. This was accomplished by repeating of student statements and encouraging students to record and share their representations and ideas on the board.
      • Maintaining continuity over time -- Ms. Nelson emphasized a sense of purpose when asking students to contribute. She directed students with phrases like, "Come up [to the board] please. Ron says that there are more diagonal lines. That Oscar didn't put enough in" (p. 181). Ms. Nelson also gave students time to understand and add clarity to other students' ideas before introducing new ideas.
      • Coordinating the collective -- Ms. Nelson actively positions students to respond to each other. When a student, Jay, had difficulty explaining an idea and Ken raised his hand, Ms. Nelson asked, "OK, do you wanna explain some more Ken? Or do you have a question for Jay?" (p. 185).
   3. Guiding the mathematics
      
      • Guiding high-level task implementation -- Ms. Nelson selected tasks that were difficult enough to invite collaboration, but guided students in ways that avoided unproductive exploration. This sometimes involved recounting the steps students had taken to reach their current thinking or requesting new representations of ideas. Either way, the focus was on how the students were thinking about the problem, and not just giving hints for the next step.
      • Guiding with a map of students' algebra learning -- This is where Ms. Nelson showed her experience with mathematics and the learning of mathematics, knowing the "pressure points" (p. 190) where students needed to pay particular attention to the structure of the mathematics.
      • Guiding by following: "going with the kids" -- Ms. Nelson showed a willingness to let go and follow students' thinking and be flexible with the intended destination of the lesson.
2. Model 2: The development of a community of collaborative learners
   
   1. The development of practices over time -- High school is a difficult time to introduce collaborative inquiry because students have longer histories with traditional mathematics and because classes meet for a limited time each day. Expectations need to be made explicit and modeled for students.
   2. The model -- Developing community is an iterative process involving tasks or strategies that Staples calls "cycle starters" (p. 196) that spur student participation, which elicits negotiation of meanings, which leads to student interpretations and understandings. From there the cycle can repeat and improve.
   3. Negotiation of meanings -- For example, early in the year students, when asked to explain, automatically assume they've given a wrong answer. Once this practice is established, students improve in the ways they respond to questions about their thinking.
      
      • Helping students make sense of practices -- Early in the year, Ms. Nelson would fill in commentary when students did work silently in front of the class, saying things like, "He is noticing a pattern over here" (p. 198). This modeled the practice of thinking aloud for the class and emphasized the value of sharing one's thinking.
      • Providing evidence for the value for learning -- When students struggled, Ms. Nelson encouraged them to stop and express what it was they were struggling with. Making mistakes became an acceptable part of doing mathematics so long as they became opportunities to learn and correct misunderstandings.
      • Negotiation of the joint enterprise -- Ms. Nelson explained early in the year that these students were doing to do mathematics differently than in the past, and that they didn't need the math dumbed-down just because they hadn't been successful before.
   4. Cycle starters -- Ms. Nelson included not only engaging tasks, but helped create a vision of how that task could be accomplished, sometimes by describing the expected collaboration but also by showing a video of older students collaborating and discussing how they worked together.
   5. Students' interpretations and understandings of practices -- From student interviews and surveys, Staples found that students interacted with each other during class either for social reasons or to make the class less boring. By the end of the year, students reported that working together helped them learn because of the opportunities to listen and respond to each others' ideas.
   6. Transforming a community's repertoire -- Ms. Nelson's effort to transform the way it does mathematics was an ongoing effort that lasted the entire year. The effort is a negotiation, where as the class built new experiences together they could reflect on what was working and adjust their practices in future lessons.

In her discussion, Staples focuses on how teachers support collaborative inquiry while maintaining their role. It is certainly possible for a teacher to ask students to share ideas or report strategies, but it takes extra effort to build that common ground where students analyze and evaluate each other's ideas. Defining this common ground is difficult and it will be different in every classroom, but it is up to the teacher to develop and maintain it throughout the school year. Teachers also must be mindful of the mathematics, even when "going with the kids." It takes a skillful teacher to subtly push the mathematics while keeping the class collaborative. Lastly, the teacher must maintain a sense for a "long-term trajectory of student learning" (p. 210), although having such a sense does not necessarily support collaboration by itself. Staples's research is thorough and well-grounded in qualitative methodology, but it is not without its criticisms. I see critiques coming from two directions: from a more cognitive perspective, Staples doesn't attend much to individual student thinking, preferring to focus on social practices and classroom norms for participation. From a more purely sociocultural perspective, Staples doesn't account for how the influence of other, beyond-the-classroom cultures and community norms affect how students approach and understand mathematics. This is not to fault Staples, however -- she prefaced her findings with defining a situative perspective, and she maintained that perspective throughout. This just means that there are multiple ways of describing classroom communities and learning, and more work can be done to describe and build bridges across multiple perspectives.

References

Boaler, J. (2008). What’s math got to do with it? How parents and teachers can help children learn to love their least favorite subject (p. 273). New York, NY: Penguin Group.

Staples, M. (2007). Supporting whole-class collaborative inquiry in a secondary mathematics classroom. Cognition and Instruction, 25(2), 161-217. doi:10.1080/07370000701301125

RYSK: Boaler's Open and Closed Mathematics: Student Experiences and Understandings (1998)

This is the sixth in a series of posts describing "Research You Should Know" (RYSK).

The math wars might have quieted a bit since their heyday in the mid- and late-1990s, but if you hold your ear up to the internet and listen closely it won't be long before you hear the sound of reformists and traditionalists trading fire. A recent story in Education News by Barry Garelick triggered a battle in the comment section, bringing out many of the usual suspects to fight for the ground held by the other side.

While I might find such battles interesting (in a "straw-men-knocked-down-per-minute" sort of way), rarely do they accomplish anything but bolster the ill-will between the two camps. Occasionally there are hints at research findings, and perhaps somebody links to another story, blog, or website, but I never see much that might convince either side they might be wrong. Remember, this is the world of mathematics we're talking about, and "proving" anything right or wrong requires a standard of evidence not easily found.

Part of what sustains the math wars is the vast divide in what the two sides see as quality research and research methods, how they see the nature of mathematics itself, and how we measure success in mathematics[1]. A lot of math warriors might be willing to concede defeat if they came out on the wrong end of a large-scale, randomized, longitudinal experiment with high-fidelity implementations of reform and traditional curriculum and pedagogy, and multiple forms of assessment measuring a range of mathematical skills and abilities. But such experiments are very rare in social science – not because nobody wants to do them, but because the randomizing and controlling of people quickly veer towards the impossible and the unethical. So in the place of an idealized experiment, researchers have been trying to answer the traditional-vs.-reform question using the best methods available.

One such researcher is Jo Boaler. If I were to conduct a Family Feud-style survey of 100 math teachers, asking them to name a math education researcher, I'd expect Boaler to make it on the board. (Debating who else would be on the list might make for interesting Google+ and Twitter fodder.) Since earning her PhD in mathematics education from Kings College, London University in 1996, she has spent time in both the U.K. and the U.S. and is currently a professor at Stanford. While perhaps not as well-known as her later Railside study, her 3-year Open and Closed case study in the U.K. of two schools with traditional and reform approaches is worth examining here.

Boaler's research grew out of a concern that mathematical knowledge, when learned in a "traditional" way (which I'll define in a moment), isn't very transferrable to contexts outside the classroom. Learning transfer is a slippery subject for learning scientists to pin down, partly because we have a history of viewing learning as a cognitive ("in the head") activity, while transfer requires us to question the importance of our surroundings to what we learn, which is often referred to as situated learning (Lave & Wenger, 1991). Boaler wanted to investigate if, and how, being taught in mathematics affected future math performance in a variety of contexts.

For Boaler's study, she spent three years in two U.K. high schools, observing the daily activities inside mathematics classrooms. A great deal of the work was ethnographic, but she also conducted about 25 interviews per year, collected about 300 surveys, and administered a series of assessments. While the schools were not chosen randomly, they were in the same community, fed by the same primary schools, and had students with very similar demographic backgrounds. Test score averages for the two schools were roughly the same for students entering the 3-year study.

Math classes in "Amber Hill," the traditional school, generally consisted of a 15-20 minute lecture and working of example problems, followed by time for students to practice similar problems. Students were tracked into one of eight different levels depending on their prior test scores and teachers' judgment of their abilities. Overall, the atmosphere was described as calm and the students were motivated; in a short study of time-on-task, Boaler never observed fewer than 90% of students doing their work during the class. However, interview and survey data revealed that students found the work to be "boring and tedious" (Boaler, 1998, p. 45), and students described math as "rule following" (p. 46) and "cue-based" (p. 47), meaning students typically expected a task to indicate which rule to follow for solving a particular type of problem.

The other school in the study, "Phoenix Park," favored progressive education over traditional schooling. The atmosphere was very relaxed, and students were encouraged to accept responsibility for their own learning. Most of the math lessons were open-ended projects and students worked in mixed-ability groups. Boaler's description of the curriculum includes tasks like, "The volume of a shape is 216, what can it be?" When students needed math they did not know, they would get help from the teacher. When students lost interest, they were free to wander both physically and mentally in search of other work that might interest them. In the same short study of time-on-task, Boaler never recorded more than 70% of students working, and some students never appeared to do any work. When asked to describe their math lessons, the most common response from students was "noisy," followed by "good atmosphere" and "interesting" (p. 50). About a fifth of the students reported not liking having so much freedom in the classroom.

When comparing student attitudes in the two schools, Boaler found that Phoenix Park students reported being more interested in their lessons/projects, while Amber Hill students complained about their textbooks. At Amber Hill, boys reported being significantly more positive about mathematics than girls; at Phoenix Park there were no such differences.

One of the ways Boaler measured math performance was to give students a pre-test measuring their skills with volume and angles, then two weeks later give them an architectural activity using those same skills in context. A score of 1 represented a correct (or nearly correct) answer, while a 2 represented an incorrect answer. The percentage of students scoring a 1 on each task is shown in the table below.

	Pre-Test Volume	Pre-Test Angle	Architectural Task Volume	Architectural Task Angle
Amber Hill	72%	94%	55%	64%
Phoenix Park	60%	94%	75%	82%

So while Amber Hill students scored better with decontextualized problems, Phoenix Park students did better with the tasks that more closely resembled using math in the real world. Boaler noticed a pattern in Amber Hill students' responses for the architectural angle task: many students took the word "angle" as a prompt to use trigonometry, even though none was needed.

While many might assume the traditional style of Amber Hill would result in those students receiving higher standardized test scores, Boaler suspected that transferring their knowledge from textbook to exam might be more difficult for Amber Hill students, as the exam contained questions that went beyond the simple application of rules and procedures. In examining GCSE exam scores from the end of Year 11, Boaler found 11% of students at both Amber Hill and Phoenix Park received an A-C grade, but 88% of Phoenix Park students passed the exam compared to only 71% at Amber Hill. Boys at Amber Hill received significantly higher grades than girls (20% to 9%), while no such significant differences were found at Phoenix Park (13% for boys, 15% for girls).

In the discussion section of the article, Boaler returns to survey data and exposes the differences in attitudes towards math among students from both schools. Amber Hill students admit that they didn't see connections between their textbook exercises and the real world, while Phoenix Park students talked more about the process of solving problems and using mathematics as an adaptable tool. In her conclusion, Boaler claims that while Amber Hill students knew more mathematics, the students from Phoenix Hill could apply more mathematics because their style of learning had forced them to become more flexible in their approach and more forgiving of their environment. Boaler does criticize the open approach of Phoenix Park because despite the relatively favorable test scores in comparison to Amber Hill, it led to a great deal of wasted student time. Regardless of the curricular and pedagogical details, Boaler's final conclusion is that "a traditional textbook approach that emphasizes computation, rules, and procedures, at the expense of depth of understanding, is disadvantageous to students, primarily because it encourages learning that is inflexible, school-bound, and of limited use" (p. 60).

Are Boaler's findings enough to end the math wars? If the answer was "yes," they would have ended in 1998 when the article was published. While the article does come across as a victory for reform, I don't think we can equate the progressive, open style of Phoenix Park with the expectation of "normal" reform classrooms. Similarly, Amber Hill might be more traditional than a "normal" traditional classroom. Still, Boaler's methodology helps shed light on how researchers try to answer the "traditional-vs.-reform" question, and this work helps us think about the importance of how we assess and how our perspective of learning changes when we view it through the lens of learning transfer. Boaler's conclusion should still be useful information regardless of your perceived approach, even if it falls short of declaring a cessation of hostilities.

[1] In addition, the lack of free public access to high-quality mathematics education research also sustains the math wars. After all, it's largely public opinion that keeps the war going, and researchers have allowed themselves to contribute to a system that discourages the public from seeing their published results. I'm hoping these posts put a dent in that knowledge gap, but I can only do so much. For a big step towards a long-term solution to this problem, I urge you to support HR 4004, the Federal Research Public Access Act, as well as http://thecostofknowledge.com/. Finally, something I think both the traditionalists and reforms can both get behind.

References



Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41-62.

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation (p. 138). Cambridge, UK: Cambridge University Press.