Showing posts with label standards. Show all posts
Showing posts with label standards. Show all posts

Coherence Gap Spreadsheet

I'm overdue in getting this out into the wider world, but I've developed a spreadsheet that incorporates all the coherence connections found in the Coherence Map and adds to that instructional time data about how many lessons and hours a math curriculum spends on each standard. Yeah, it's a lot. But as I talked to math teachers and leaders at the end of last spring, I felt there were a lack of tools available that incorporated both coherence and instructional time, and my solution was a big Excel workbook with lots and lots of rows of lesson data, some long VLOOKUP formulas, and some conditional formatting to make the results readable. If you geek out over spreadsheets and math curriculum planning, I think you'll like it.

The lesson data comes from EngageNY for K-5 and Illustrative Mathematics for Grades 6 through Algebra 2. I didn't have any special preference for one set of curriculum materials over any other (and neither does the State of Colorado), but both of these are open educational resources with lesson alignments and time estimates, so I used them. You can substitute time data in for whatever materials you'd like, but be warned that you're looking at 3500+ rows of data, so either bring a team of people with you or learn to write some code that scrapes data from publishers' websites and formats it for you. (I chose the latter.)

With the help of a pivot table and some lookup functions, what this spreadsheet allows you to do is indicate some percentage of coverage you thought each standard has gotten (if making sense of past curriculum decisions) or will get (if planning for future curriculum decisions). In return, the spreadsheet reports back to you how much instruction for each standard is left unfinished, and how much future instruction (following arrows in the Coherence Map) might be at risk. Just be mindful that the spreadsheet is like a lot of models—wrong, but possibly useful. The instructional time estimates might be flawed, not everything aligns as neatly as I'd like, the data may not really reflect your materials or pacing, and the crude way the coverage formulas work might just be wrong. So if it gives you some data that you just know is flawed, then maybe it is. But if you're patient with it, and you aren't afraid to look beyond the conditional formatting color scheme and dig more deeply into why instructional time is allocated where it is, then I think this spreadsheet can give you something to work with as you make decisions about your curriculum planning.

This was one of my on-the-job projects as math specialist for the Colorado Department of Education, so CDE is hosting the spreadsheet itself. Head on over to the Coherence Gap Spreadsheet page to download the latest version of the file. I also urge you to watch the tutorial video, which I'll also embed here. I kept it as short as I could at 12 minutes, and if you're already familiar with the Coherence Map you can skip the first 2:00.

If you have any questions about the spreadsheet, please let me know. And if you modify the spreadsheet to make it better or more inclusive of more curriculum materials, I'd really like to know about that, too.

Starting the Standards Era: NCTM and the 1980s (Part 6 of 6, Focusing the Council on Standards)

(See Part 1, Part 2, Part 3, Part 4, and Part 5 of this six-part series.)

The successful release of the 1989 NCTM Standards paved the way for the release of the next two NCTM standards documents, the Professional Standards for Teaching Mathematics (1991) and the Assessment Standards for School Mathematics (1995). While neither received all the attention of the 1989 Standards, a change in administration in the federal government and changing attitudes at private foundations meant money for later Standards-based projects was more easily obtainable.

In order to provide teachers and district-level mathematics specialists a clearer vision of what Standards-guided lessons would look like, the NCTM launched a project called the Addenda Series, with a committee chaired by Bonnie Litwiller of the University of Northern Iowa1. Although initially intended to produce just a few books a year for one or two years, the project eventually produced 22 books in five years, covering all grade levels K-12. Each book in the Addenda Series provided a set of lesson plans that a teacher could use directly in his or her classroom, offering some of the specificity lacking in the original Standards. The Addenda Series also supported NCTM financially, as it became their most profitable set of publications (S. Frye, personal communication, April 19, 2013).

In order to focus all the NCTM publications on the Standards, a deliberate effort was made by the editors of NCTM's journals to acquire and publish articles that cited the Standards (Lindquist, 2003, p. 837). Authors of research articles that did not refer to the Standards were asked as part of the peer review process to refocus their writing to include the Standards. With this de-facto policy in place, soon nearly every article listed the Standards as a reference. Somewhat ironically, the Standards themselves contain a reference list of only 27 sources (NCTM, 1989, pp. 257-258).

Conclusion

The creation and publication of the NCTM Standards is generally recognized as the event that launched our current era of standards-based reform. Given the rapidity with which educational reforms come and go, such a lasting impact from a document published almost 25 years ago deserves to be well-understood by education policymakers as well as teachers and other education stakeholders. The most significant positive, negative, and fortunate aspects of NCTM's Standards process can be summarized as:

Positive:

  • Leadership desired an organization-level policy influence.
  • Working groups possessed expertise and represented diverse stakeholders.
  • Goals were set conservatively in an effort to broaden public acceptance.
  • Drafts of the Standards were sent to a very wide audience for review and commentary.
  • Standards were promoted through a massive public relations campaign.

Negative:

  • Despite seeking consensus, reconciliation with the most vocal critics in the mathematics community has yet to happen.
  • The working groups lacked writing talent.

Fortunate:

  • A well-timed "crisis" came in the form of A Nation at Risk.
  • NCTM membership rebounded in the mid-80s before the Standards project had an opportunity to put the organization in greater financial jeopardy.
  • Attitudes about the federal government's role in education, as well as national efforts like the Standards, became more favorable after the end of the Reagan Administration.

It's evident that NCTM's leadership in standards-based educational reform didn't come without a sizeable bit of good fortune. The shifting of any number of events by a year or two might have jeopardized the entire process, or relegated the Standards to be that "book on the shelf" to which few paid much attention.

When compared to the Common Core State Standards, a few significant differences stand out to me. First, the NCTM Standards were created largely for the purposes of comparing and judging curriculum, whereas the CCSSM were created as student learning targets and as part of a larger accountability structure. The NCTM Standards were not grade-level specific like the CCSSM, nor were they ever "adopted" wholesale by states or districts. Instead, the NCTM Standards became a foundation for states and districts to write their own standards, and the CCSSM represents the effort to de-duplicate the efforts of states by having a single, agreed-upon set of standards. Although not without their detractors, standards efforts on this scale do have the potential to drive positive change and anchor collaboration between educators across states and districts. Time will tell if any lasting effects of the CCSSM measure up to those of the NCTM Standards, and how.

References

Lindquist, M. M. (2003). My perspective on the NCTM Standards. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 819-842). Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics (p. 196). Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics (p. 102). Reston, VA: Author.


  1. Bonnie Litwiller was my academic advisor and twice my methods professor while I was an undergraduate mathematics major at UNI. 

Starting the Standards Era: NCTM and the 1980s (Part 5 of 6, Making a Draft Widely Available for Review; Publishing and Promoting)

(See Part 1, Part 2, Part 3, and Part 4 of this six-part series.)

So far this series has described the first three of six characteristics of the policy process I outlined in Part 1. This installment will look at the next two characteristics. Despite their shorter descriptions, both were critical in NCTM's effort to have the Standards see wide adoption.

Making a Draft Widely Available for Review

There was one aspect of the Standards draft review process that had significant policy implications. Instead of just sending drafts to a limited number of outside experts, as is often the norm in such cases, NCTM sent out 10,000 copies of the 1987 draft to more than fifty other groups with interests in mathematics education (McLeod, 2003, p. 779). Every single page of the draft contained room for comments, and the working groups received comments by the thousands. Not only did such wide distribution create anticipation for the final draft, the NCTM garnered the support of sixty organizations whose names were printed in the opening pages of the final draft (NCTM, 1989, pp. vi-viii). Listing as endorsers the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Mathematical Sciences Education Board helped moderate the opinion by some that mathematicians were excluded from the Standards writing process.

In the end, the Standards incorporated the perspectives of many people and organizations, but not without compromise. Building consensus while being provocative is a tricky balance, something to which Michael Apple (1992) applied the term "slogan system," meaning they were

a statement of goals that was specific enough to provide direction to the field, vague enough to be acceptable to most mathematics teachers, and novel enough for its vision to catch the attention of the many different groups having a stake in mathematics education. (McLeod, 2003, p. 783)


Publishing and Promoting

While the public relations campaign undertaken by the NCTM to promote the Standards may not have been notable from Mary Lindquist's perspective as a writer (see the difference between her four characteristics and my six in Part 1), it certainly deserves attention as a matter of policy. Without a massive effort, the immediate and lasting policy influence of both the NCTM and the Standards would have certainly been reduced. By the time of publication in March 1989, the total expense of the Standards project had reached approximately $1,000,000, far exceeding the initial estimate of $258,000 (McLeod et al., 1996, p. 44). Included in the million-dollar total was $200,000 in expenses paid to public relations firms. Without this and continuing effort, the worry was that the Standards would be resigned to "sit on shelves" (Lindquist, 2003, p. 840), where all but a few curious graduate students would ever look at them again.

The public relations efforts had all the signs of a six-figure expense (McLeod et al, 1996, pp. 15-16). First, NCTM leadership, including President Shirley Frye and Tom Romberg, were coached to improve their ability to positively present themselves and to handle tough questions gracefully. They then hosted a press conference in Washington D.C. for about 200 members of the media. NCTM leaders made appearances on the Today Show and other major news programs and Astronaut Sally Ride was brought in to help by lending her endorsement. A video featuring jazz musician Wynton Marsalis describing the Standards was "shown over 6000 times by 121 television stations, reaching an audience in the millions" (McLeod et al., 1996, p. 64).

Perhaps most significant was how many copies of the Standards the NCTM had arranged to give away. Unlike the Agenda's relatively short 30 pages, the Standards were 258 pages in length. Still, the NCTM gave away a copy to each one of their 51,000-plus members, as well as anyone and everyone who might have influence but wasn’t an NCTM member. Judith Sowder, Standards Coordinating Committee chair, remembered:

The mailing lists were enormous. The NCTM lobbyist took [the Standards] around personally and handed them to members of Congress. Certainly every dean of sciences, every chair of a mathematics department, every math coordinator, high school principal, and elementary school principal who was on our mailing lists got one. We sent to PTA presidents, school board presidents, and on and on and on. Every mailing list that could possibly be used was used. (McLeod et al., 1996, p. 63)

While the size of this giveaway represented a huge cost to NCTM, it was necessary to ensure widespread adoption. Fortunately for NCTM and their budget, by 1995 more than 258,000 copies of the Standards had been distributed, including the giveaways, and the $25 cost per purchased copy made up for the lost revenue and helped pay for the expenses of the project (McLeod et al., 1996, p. 63).

Now that NCTM had written, published, and promoted the Standards, the last important piece was to make sure they played a part in future efforts. In Part 6, we'll look at how the NCTM focused efforts around the Standards, and I'll wrap up the series with some reflection.

References

Apple, M. W. (1992). Do the standards go far enough? Power, policy, and practice in mathematics education. Journal for Research in Mathematics Education, 23(5), 412-431. doi:10.2307/749562

Lindquist, M. M. (2003). My perspective on the NCTM Standards. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 819-842). Reston, VA: National Council of Teachers of Mathematics.

McLeod, D. B. (2003). From consensus to controversy: The story of the NCTM Standards. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 753-818). Reston, VA: National Council of Teachers of Mathematics.

McLeod, D. B., Stake, R. E., Schappelle, B. P., Mellissinos, M., & Gierl, M. J. (1996). Setting the standards: NCTM's role in the reform of mathematics education. In S. A. Raizen & E. D. Britton (Eds.), Bold ventures: Case studies of U.S. innovations in mathematics education (pp. 13-132). Dordrecht, The Netherlands: Kluwer.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Starting the Standards Era: NCTM and the 1980s (Part 4 of 6, Establishing and Supporting Working Groups)

(See Part 1, Part 2, and Part 3 of this six-part series.)

NCTM's standards-writing process began in 1986 when the Council proposed the creation of a Commission on Standards, chaired by Tom Romberg, and four working groups: grades K-4, 5-8, 9-12, and evaluation. Each working group was chosen for its expertise and consisted of six people, generally a mix of mathematics teachers, state and district math supervisors, mathematics professors from major university mathematics departments, and mathematics education researchers (Lindquist, 2003, pp. 826-827; McLeod, 2003, pp. 772-773). The working groups were somewhat conservative, as radical suggestions would likely make for a less marketable policy recommendation. John Dossey, then president of NCTM, remarked that each group included

somebody who had been around and had a lot of experience – who could represent not a traditional view, but someone who understood the status quo well, who understood the dangers of change, and who was a worker for change, but who knew that you could not just flip a switch and have it happen. (McLeod et al., 1996, p. 46)

While expertise and diversity are generally key ingredients in a policy-making process, a quality that may have been overlooked was the recruitment of quality writers. While some members of the working groups had written textbooks, "neither writing experience nor the ability to produce polished prose was a criterion for selection, and the writing teams often struggled to produce high-quality text" (McLeod, 2003, p. 774). Writing quality was an unexpected struggle during the almost two-year process of writing the Standards.

Although the working groups were tasked with writing standards focusing on mathematical content, they were also mindful of equity issues – including how the inclusion of equity statements might help or hinder the adoption of the Standards. Christian Hirsch, chair of the 9-12 working group, remarked:

I think a careful look at the Standards would show that, in the case of the high school mathematics curriculum, there were two issues that the Standards politically decided not to take a stand on. One was the issue of tracking, and the other was the issue of whether the mathematics studied each year at the high school should be an integrated or unified curriculum, as opposed to a curriculum that was subject-matter oriented each year: algebra, geometry, advanced algebra. That decision was very conscious, in that we felt that we needed to identify in the Standards what we believed at the time in history to be the most important mathematics that all students should have the opportunity to study. And that in itself was advancing thinking on the curriculum quite a ways, because if one looked at the curriculum of the 1970s and 1980s, there was a marked contrast between the mathematics that was in college prep programs and the mathematics that one found in general math, consumer math, remedial courses. We felt it was most important to get out on the table (and over time gain acceptance for) the notion that all kids should be studying different mathematics, rather than getting the Standards caught up in a heated debate over how that mathematics could be organized and made available to students – that is, through sequences of courses that may or may not be tracked. (McLeod et al., 1996, pp. 56-57)

With the exception of a small grant from the AT&T Foundation for $25,000, NCTM chose to finance the writing of the Standards themselves, despite having recently been in significant financial difficulty. The organization had seen its membership fall from 82,000 in 1968 to 56,000 in 19831, and the loss of revenue forced the Board of Directors to consider a proposal to eliminate NCTM's publication program (McLeod et al., 1996, p. 20). Despite the risk of bearing the responsibility for the Standards total estimated cost of $258,000 (McLeod et al., 1996, p. 42) former Executive Director James Gates claimed "the proposal [to fund the Standards] was not submitted to either NSF or the U.S. Department of Education, so that no claims could be made that the federal government had funded the development of curriculum and evaluation standards" (Gates, 2003, p. 742). In addition, the self-funding of the Standards and the decision to not write textbooks, as had been the case during the new math era, afforded the working groups relative independence from textbook publishers. The "corrupting process" (McLeod et al., 1996, p. 33) of working with textbook publishers was a shared concern among the working groups, explained by Arthur Coxford in his chapter in A History of School Mathematics:

Publishers tend to be concerned with the 'bottom line,' whereas curriculum developers desire to try new ideas and organizations. Editors for publishers listen carefully to state textbook adoption committees and to teachers in the field. Neither of these groups was demanding radically different curricula in the 1980s. In fact, they often recommended retaining topics (Cramer's rule or computation using logarithms, for example) long after the usefulness, mathematical or in application, of the topic had diminished. Often it seemed such recommendations were based on an individual's opinion rather than the result of a careful analysis of needs. (Coxford, 2003, p. 613)

While self-funding did afford the working groups a degree of independence and James Gates' statement is at least partially true, the reality of the situation is that the federal government had very little, if any, money to give for a project like the Standards. In 1982, the Reagan Administration has stripped all K-12 funding for mathematics and science from NSF's budget (McLeod et al., 1996, p. 25). Moreover, the same Reagan Administration that had recently sought to dismantle the U.S. Department of Education in the name of local control was not likely to award large sums of money for the development of a national set of curriculum standards. NCTM had applied for a sizable amount of other private money, but the AT&T grant was the only one awarded. Clearly the organization had no other real options but to pay for the Standards itself and use the independence to its advantage, including spinning the effect of that independence as a policy tool.

In Part 5 of this series, we'll look at how NCTM collected and incorporated feedback about the Standards and the measures they took to promote the published draft.

References

Coxford, A. F. (2003). Mathematics curriculum reform: A personal view. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 599-621). Reston, VA: National Council of Teachers of Mathematics.

Gates, J. D. (2003). Perspective on the recent history of the National Council of Teachers of Mathematics. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 737-752). Reston, VA: National Council of Teachers of Mathematics.

Lindquist, M. M. (2003). My perspective on the NCTM Standards. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 819-842). Reston, VA: National Council of Teachers of Mathematics.

McLeod, D. B. (2003). From consensus to controversy: The story of the NCTM Standards. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 753-818). Reston, VA: National Council of Teachers of Mathematics.

McLeod, D. B., Stake, R. E., Schappelle, B. P., Mellissinos, M., & Gierl, M. J. (1996). Setting the standards: NCTM's role in the reform of mathematics education. In S. A. Raizen & E. D. Britton (Eds.), Bold ventures: Case studies of U.S. innovations in mathematics education (pp. 13-132). Dordrecht, The Netherlands: Kluwer.

National Council of Teachers of Mathematics. (2013). NCTM at a glance. Retrieved from http://www.nctm.org/about/content.aspx?id=174


  1. Membership began to rebound in 1984. NCTM's membership grew to 118,000 in 1995 (McLeod et al., 1996, p. 20) and currently stands at 80,000 members (National Council of Teachers of Mathematics, 2013). 

Starting the Standards Era: NCTM and the 1980s (Part 3 of 6, Meeting a [Perceived] Need)

(See Part 1 and Part 2 of this six-part series.)

"Exactly where the Agenda's call for action might have led without the appearance of a new crisis is not clear" (Fey & Graeber, 2003, p. 553). If the research and public sentiment regarding mathematics education had looked positive as the country moved into the 1980s, there would have been little need for the NCTM to flex its new policy muscles. But that wasn't the case. Instead, growing concern over the state of math education would give the NCTM a reason to put the Agenda for Action into action.

In the late 1970s, the NSF funded a series of surveys and case studies to determine a baseline of the nation's mathematics performance. The case studies indicated that most classrooms were still exhibiting a traditional view of mathematics and showed little influence of the new math efforts of the 1960s (McLeod, 2003, p. 757). Furthermore, early results from the National Assessment of Educational Progress (NAEP) raised doubts that students were able to perform anything but the most basic mathematical tasks.

Less publicly visible but yet of concern to mathematics educators was the continued trend towards "basic" math textbooks. In particular, the claims of outspoken textbook author John Saxon "became a preoccupation of NCTM leaders" (McLeod, 2003, pp. 760-761). Saxon (1982), in a three-page Phi Delta Kappan article, made boisterous claims about the effectiveness of his textbooks. The article lacked a description of how (or if) the treatment and control groups were randomized, what textbooks were used by the students in the control group, how the assessment used to measure students' learning was constructed, and failed to use any real statistical tests. It did, however, include the address of the publisher and the cost of his textbook, as well as statements like, "A general scanning of the scores suggests that gifted students who used the normal textbooks were severely damaged and that less gifted students who used the normal textbooks were destroyed" (p. 484).

NCTM’s Research Advisory Committee (RAC) fielded concerns over Saxon's claims, some requesting censure of Saxon's texts and others requesting further research regarding the effectiveness of the Saxon texts. John Dossey, NCTM president from 1986-1988, recalled that "RAC members felt that it was inappropriate for professional groups to censure material, especially in the absence of an agreed-upon set of standards" (McLeod et al., 1996, p. 31). Concurrently, NCTM's Instructional Issues Advisory Committee (IIAC) was considering the creation of a document that could be used by schools when selecting textbooks. Jim Fey, an IIAC member at the time, said, "There was some concern from several places that textbooks, and therefore curricula, were being driven by non-professional considerations, political log rolling, and so on" (McLeod et al., 1996, p. 31). The RAC and IIAC were already considering such a textbook selection document in the spring of 1983 when a much more public educational crisis would demand the attention of the NCTM.

In April the National Commission on Excellence in Education (1983) published A Nation at Risk: The Imperative for Educational Reform. This critical document used Cold War-era language combined with threats of losing our nation's economic competitiveness to assert that it was imperative that schools change to meet the nation's growing needs. A Nation at Risk convinced many that an increase in the amount rigorous coursework required in schools, specifically in mathematics and science, should be a top national priority. While there is substantial evidence suggesting that the nation wasn't any more "at risk" than it ever had been (Berliner & Biddle, 1995), the perception of risk was more important than the truth.

By the end of 1983, two small conferences were held to determine the math education community's response to A Nation at Risk. Only sixty-eight people attended in total, with only six people attending both conferences (McLeod, 2003, p. 767). One of those six people was Tom Romberg, the University of Wisconsin professor who would later be named chairperson of the NCTM Standards Commission. Among the recommendations to come out of those conferences was the organization of a group who could write a set of guidelines specifying qualities of a proper mathematics curriculum (Romberg & Stewart, 1984).

Romberg remembered that "A Nation at Risk served primarily as a spark plug, a starting point for people" (McLeod et al., 1996, p. 27). Others downplayed the influence of A Nation at Risk. Mary Lindquist claimed "The Standards came mainly from within mathematics education rather than as a reaction to A Nation at Risk or federal policies" (McLeod et al., 1996, p. 37). The deciding measure of A Nation at Risk's impact might be found in the Standards themselves, in the first line of the first paragraph of the Introduction: "These standards are one facet of the mathematics education community's response to the call for reform in the teaching and learning of mathematics" (NCTM, 1989, p. 1). The footnote for that sentence contains the statement "See A Nation at Risk."

NCTM had prepared itself to take a stand on matters of policy and now they had their greatest opportunity. In Part 4 of this series, we'll look at how NCTM organized itself to write the Standards, and the risks they took and avoided in doing so.

References

Berliner, D. C., & Biddle, B. J. (1995). The manufactured crisis: Myths, fraud, and the attack on America’s public schools (p. 414). New York, NY: Basic Books.

Fey, J. T., & Graeber, A. O. (2003). From the New Math to the Agenda for Action. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (Vol. 1, pp. 521-558). Reston, VA: National Council of Teachers of Mathematics.

McLeod, D. B. (2003). From consensus to controversy: The story of the NCTM Standards. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 753-818). Reston, VA: National Council of Teachers of Mathematics.

McLeod, D. B., Stake, R. E., Schappelle, B. P., Mellissinos, M., & Gierl, M. J. (1996). Setting the standards: NCTM‟s role in the reform of mathematics education. In S. A. Raizen & E. D. Britton (Eds.), Bold ventures: Case studies of U.S. innovations in mathematics education (pp. 13-132). Dordrecht, The Netherlands: Kluwer.

National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform. Washington, D.C. Retrieved from http://www2.ed.gov/pubs/NatAtRisk/index.html

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Romberg, T. A., & Stewart, D. M. (Eds.). (1984). School mathematics: Options for the 1990s. Retrieved from http://www.eric.ed.gov/ERICWebPortal/detail?accno=ED250196

Saxon, J. (1982). Incremental development: A breakthrough in mathematics. Phi Delta Kappan, 63(7), 482-484. Retrieved from http://www.jstor.org/stable/20386409.

Starting the Standards Era: NCTM and the 1980s (Part 2 of 6, Asserting a Policy-Minded Orientation)

From the NCTM's inception in 1920 until the 1960s, the organization "played an important but usually secondary role" (McLeod, Stake, Schappelle, Mellissinos, & Gierl, 1996, p. 18) in mathematics education policy. NCTM's primary role "focused on supporting mathematics teachers through the exchange and promotion of good ideas, not through its influence on educational policy" (McLeod, 2003, p. 759), and many NCTM leaders thought it was best to avoid "positions that might be opposed by some of its members" (McLeod et al., 1996, pp. 18-19). Therefore, during the Sputnik-era calls for reform in the late 1950s and the "new math" era of the 1960s (Fey & Graeber, 2003), organizational leadership from NCTM was insignificant.

Fueled by the battles over the new math and events of the mid-1960s, attitudes at NCTM began to change:

Although the attempt to change school mathematics during the new math era was not very successful, the idea that an activist professional organization could have an impact on society still had some appeal. The change from passive to more active stances was a topic of discussion for many professional organizations during the 1960s and 1970s. Opposition to the war in Vietnam was one source of these discussions. As academics became involved in teach-ins and other protests against the war, there was a natural spread of their concerns into the domain of professional organizations. (McLeod, 2003, p. 758)

NCTM's hands-off policy stance changed in 1966 when the Board of Directors voted to be more willing to assert its position on controversial issues. Reflecting on his thirty-one years (1964-1995) as NCTM Executive Director, James D. Gates (2003) characterized the decision and its effects: "It was a bold step for the Council, to take actions that were more visible in the public sector, leading to the development and distribution of position statements, the publication of guidelines and standards, and testimony before congressional committees" (p. 747).

While the NCTM struggled to use its new policy-minded powers during the 1970s (Fey & Graeber, 2003), the critical turning point came with the election of Shirley Hill as NCTM President in 1978. Joe Crosswhite, NCTM president from 1984-1986, remarked that, "Prior to Shirley's time, you couldn't interest an NCTM president in having a national presence in Washington – an NCTM presence" (McLeod et al., 1996, p. 19). Shirley Hill explained that she

felt a certain frustration that we weren't being listened to seriously enough outside our own circles....I remember attending some meeting of the presidents of like organizations in Washington, DC, in the 1970s and noticing the frequent absence of the president of one of our sister organizations. It turned out that he was being escorted by his staff government relations expert in visits to members of Congress. At that time his organization seemed to be very influential in the establishment of federal programs. I thought that we in NCTM should be doing more of these things. I thought that we and most of our sister organizations were being a little naïve about government relations and public relations at that time. (McLeod et al., 1996, pp. 19-20)

In addition to hiring Richard Long, a former lobbyist for the International Reading Association (McLeod, 2003, p. 760), two documents published by NCTM during this time mark NCTM's emerging policy perspective. The first was actually a republishing of a position paper of the National Council of Supervisors of Mathematics (NCSM), a sister organization of the NCTM. The paper, A Position Paper on Basic Mathematical Skills (1977), was notable because instead of refuting the "back to basics" theme of school mathematics in the 1970s, it co-opted the language and redefined the meaning of "basic skills" for NCSM's and NCTM's own purpose (Fey & Graeber, 2003, p. 552; McLeod, 2003, p. 761). With this action, NCTM and NCSM showed that both organizations understood the importance of controlling the vocabulary and discourse in educational policy.

The second document published by the NCTM solidified their stance as a policy influencer. The Agenda for Action (1980) was the product of NCTM's Committee for Mathematics Curriculum for the 1980s, chaired by George Immerzeel of the University of Northern Iowa. While only about thirty pages in length and containing eight somewhat non-specific recommendations, the Agenda was NCTM's most prominent and powerful policy document to date, and "laid the groundwork for a major reform effort that continued through the end of the twentieth century" (Gates, 2003, p. 741). Shirley Hill described the context for the Agenda at her 1980 presidential address:

In the 1960s we learned that curriculum change is not a simple matter of devising, trying out, and proposing new programs. In the 1970s we learned that many pressures, from both inside and particularly outside the institution of the school, determine goals and directions and programs....A major obligation of a professional organization such as ours is to present our best knowledgeable advice on what the goals and objectives of mathematics education ought to be....In my opinion, we are approaching a crisis stage in school mathematics. Policy makers in education are not confronting the deepest problems because the public and its representatives have been diverted by a fixation on test scores....We are still battling an excessive narrowing of the curriculum in the name of "back to basics." (Hill, 1980, pp. 473-476, as cited in McLeod et al., 1996, pp. 24-25)

Furthermore, in the introduction of the 1983 NCTM Yearbook, The Agenda in Action, Shirley Hill described the NCTM's implementation of the Agenda in five categories:

  1. Public relations.
  2. Political action.
  3. Support for local efforts.
  4. Collection and dissemination of model programs.
  5. Production of guidelines and instructional resources.

Certainly the first two items in the list would have been far less likely to appear even ten years earlier. The words and actions of Shirley Hill clearly demonstrate the policy orientation NCTM had asserted by the early 1980s. But a willingness to affect policy and an opportunity to affect policy are two different things, and that opportunity would come soon enough. In Part 3 of this series, we'll look at the events that set NCTM to work on the Standards.

References

Fey, J. T., & Graeber, A. O. (2003). From the New Math to the Agenda for Action. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (Vol. 1, pp. 521-558). Reston, VA: National Council of Teachers of Mathematics.

Gates, J. D. (2003). Perspective on the recent history of the National Council of Teachers of Mathematics. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 737-752). Reston, VA: National Council of Teachers of Mathematics.

Hill, S. (1983). An agenda for action: Status and impact. In G. Shufelt & J. R. Smart (Eds.), The
agenda in action
(pp. 1-7). Reston, VA: National Council of Teachers of Mathematics.

McLeod, D. B. (2003). From consensus to controversy: The story of the NCTM Standards. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 753-818). Reston, VA: National Council of Teachers of Mathematics.

McLeod, D. B., Stake, R. E., Schappelle, B. P., Mellissinos, M., & Gierl, M. J. (1996). Setting the standards: NCTM's role in the reform of mathematics education. In S. A. Raizen & E. D. Britton (Eds.), Bold ventures: Case studies of U.S. innovations in mathematics education (pp. 13-132). Dordrecht, The Netherlands: Kluwer.

National Council of Supervisors of Mathematics. (1977). Position paper on basic mathematical skills (p. 4). Minneapolis, MN. Retrieved from http://www.eric.ed.gov/ERICWebPortal/detail?accno=ED139654

National Council of Teachers of Mathematics. (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, VA. Retrieved from http://www.nctm.org/standards/content.aspx?id=17278

Starting the Standards Era: NCTM and the 1980s (Part 1 of 6, Where Curriculum and Policy Meet)

The Common Core State Standards might be the current story, but to gain a broader perspective of this "standards era" of educational reform we would be wise to look at where the era got its start. Over the next six posts, borrowing generously from the work of Douglas McLeod and others, I'll attempt to tell the story of how the National Council of Teachers of Mathematics (NCTM) created the first widely-recognized set of curriculum standards, and the policy process that developed along the way.

NCTM's publication of the Curriculum and Evaluation Standards for School Mathematics (1989) stands as the landmark event that launched our nation's current era of standards-based educational reform1. While the effect of this effort led to a new period of reform in school mathematics, marked by new textbooks and materials and what has come to be known as the "math wars," the process undertaken by NCTM in writing the Standards has been both praised and panned by critics. Diane Ravitch (1995) claimed the Standards "emerged from a successful consensus process that included many classroom teachers and the nation's leading mathematics educators.... [They are] an example for emulation" (p. 57). Roy Romer (1995), then the Governor of Colorado, said the Standards "were arrived at correctly, from the bottom up. They represent the best thinking in the country, collectively" (p. 67).

Critics were far tougher on NCTM and the process for writing standards for school mathematics. Ralph A. Raimi, a prominent figure in the math wars, claimed that whenever he was asked to help write or review math standards, he'd send the following recommendation:

If your standards were composed without the significant participation of mathematicians, let me advise you to go down to your best state university and find a professor of mathematics, at least 40 years old, who is willing to help you. He need not have heard of Piaget and Bruner, and he might very well be of such a personality that you would never trust him in a fifth grade class, but he should be an English-speaking American who himself has gone through our public school system, and he should be a genuine mathematician who has published at least a handful of research articles in the refereed professional journals of pure or applied mathematics. (Not journals of math education; you have such people in your department of education already.) Find out that this mathematician is willing to devote a few days to your project. Give him a copy of the Fordham Foundation report on the state standards to read....Then give him a copy of your own state's draft standards and ask for written commentary. Then use it. (Raimi, 2000, p. 57)

Raimi's suggested process for writing or reviewing mathematics standards might have some admirers, but it does not reflect the process undertaken by most standards-writing groups who wish to have a lasting impact on education practice and policy. Because the NCTM Standards have had an influence lasting now over twenty years, this series of writings will examine the specific process undertaken by NCTM that led to the publication of the Standards in 1989.

The Standards as a Policy Process

A review of the literature describing the NCTM's efforts can be undertaken from multiple perspectives. Teachers of mathematics might be most interested in the content of the Standards themselves, along with the contrasting arguments that influenced the curricular content emphasized and de-emphasized by the Standards. Historians of education might wish to study the development of the Standards as a sequence of events set in the context of greater educational and societal movements. Those who participated in the writing of the Standards bring yet another perspective of the process, such as that of Mary Lindquist, a member of the Grades K-4 working group that wrote the Standards. In Lindquist's chapter of NCTM's A History of School Mathematics (2003), she describes the effort to develop and promote the standards as having "four fundamental characteristics:"

  1. Accepting responsibility for standards.
  2. Establishing and supporting working groups.
  3. Making a draft widely available for review.
  4. Focusing the Council on standards.

Although any review of the NCTM's standards-writing process will probably be more alike than different, to gain a broader, policy-making perspective, this series of posts will be organized into six areas that share much similarity with Mary Lindquist's fundamental characteristics:

  1. Asserting a policy-minded orientation.
  2. Meeting a (perceived) need.
  3. Establishing and supporting working groups.
  4. Making a draft widely available for review.
  5. Publishing and promoting.
  6. Focusing the Council on standards.

The addition of the first two areas acknowledges that efforts to impact policy are: (a) generally conscious efforts undertaken by an organization and (b) most successful when done in response to a perceived crisis. The fifth area, publishing and promoting, describes the sometimes extraordinary effort an organization must take to make their message heard and to make it lasting.

In Part 2, we'll look at how and why NCTM decided to assert itself in the education policy arena.

References

Lindquist, M. M. (2003). My perspective on the NCTM Standards. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (pp. 819-842). Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics (p. 196). Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics (p. 102). Reston, VA: Author.

Raimi, R. A. (2000). Judging state standards for K-12 mathematics education. In S. Stotsky (Ed.), What’s at stake in the K-12 standards wars (pp. 33-58). New York, NY: Peter Lang.

Ravitch, D. (1995). National standards in American education (p. 223). Washington, D.C.: Brookings Institution.

Romer, R. (1995). Explaining standards to the public. In D. Ravitch (Ed.), Debating the future of American education: Do we need national standards and assessments? (pp. 66-72). Washington, D.C.: Brookings Institution.


  1. Although NCTM’s Curriculum and Evaluation Standards for School Mathematics is just the first in a set of three publications, with the second two addressing the teaching (NCTM, 1991) and assessment (NCTM, 1995) of mathematics, this first document is the most well-known and is frequently referred to simply as the Standards

Common Core Conundrum: Absolute Value

I should begin by first disclaiming that while I'm generally pro-standards, I'm also somewhat agnostic about standards. How can that be? Without going into much detail, I believe (a) what we "count" as mathematics is socially determined and standards documents are just part of that determination, (b) I will never perfectly agree with a standards document, but the amount of agreement should not be underestimated, and (c) if there's any power to standards, it's in how they're implemented -- and "good implementation" is very likely to be seen as just "good teaching" under a different set of standards, or no standards at all. It is with this mindset that I watch with some amusement (and sometimes, disappointment) arguments against the Common Core State Standards because of some poorly designed accountability measure. I'm pretty sure poorly designed accountability measures would be a concern right now regardless of the standards in place.

Still, I occasionally see things in the Common Core math standards (CCSSM) that make me stop and wonder, "How are teachers going to deal with this?" One recent instance of that was with how the CCSSM addresses the topic of absolute value. As I looked through Discovering Algebra: An Investigative Approach, I saw the typical "distance from zero" definition followed by an investigation that included this rather unhelpful picture and caption:

(Clearly portrays Elvis? The middle guy looks like Matthew Perry joined a Vegas lion-taming act.)

From there, Discovering Algebra presents students with equations to solve, such as:

$$ | x - 2| + 7 = 12 $$

But do the CCSSM call for solving these kinds of equations? I went searching through the CCSSM for all mentions I could find of absolute value, and here's what I found.

Grade 6


CCSS.Math.Content.6.NS.C.7 Understand ordering and absolute value of rational numbers.
CCSS.Math.Content.6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
CCSS.Math.Content.6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that –3oC is warmer than –7 oC.
CCSS.Math.Content.6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
CCSS.Math.Content.6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
CCSS.Math.Content.6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

and

CCSS.Math.Content.6.SP.B.5c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

If all goes according to plan, there shouldn't be a need to define absolute value in 8th or 9th grade algebra, as it will already have been introduced in 6th grade. I know there are concerns about content being "developmentally inappropriate" for lower grade levels in the CCSSM, but without any personal data to the contrary, I imagine students could come to understand absolute value along with understanding positive and negative numbers. The last standard above, from the moment I first saw it, has been a point of fascination for me. How will 6th grade teachers help students learn topics like interquartile range and mean absolute deviation? You need absolute value for mean absolute deviation, but I don't think that will be the hangup for those who struggle with that standard.

So where else is absolute value mentioned?

Grade 7


CCSS.Math.Content.7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

and

CCSS.Math.Content.7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

Now we have absolute value as it relates to subtraction, and we get another mention of mean absolute deviation in the 7th grade statistics and probability standards. There's nothing yet about solving equations with absolute value or graphing absolute value functions.

Grade 8


Nothing. No mention of absolute value or mean absolute deviation.

High School


The CCSSM doesn't specify what specifically belongs in 9th grade versus other grades, but here's what I found across the whole of the HS CCSSM standards.

CCSS.Math.Content.HSN-VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

CCSS.Math.Content.HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

CCSS.Math.Content.HSF-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

The first of the three standards above, I think we can agree, goes well beyond what we'd expect to find in a traditional Algebra 1 class. The other two standards address the graphing of various kinds of functions, and depending on the activity might belong in either a traditional Algebra 1 or Algebra 2 course, or more likely, both. But do you notice what's not there? Solving equations with absolute value functions. Sorry, \( | x - 2 | + 7 = 12 \), but it looks like you didn't get invited to the CCSSM party. Even if you interpret the second standard to include solving absolute value equations, it only asks for an approximation or to use technology.

Conundrum

The researcher in me wonders what Algebra 1 teachers will do when they get to a lesson on absolute value. Will some just teach it out of habit? Will some think it was excluded due to a CCSSM oversight? Will some modify the lesson to emphasize the graphing of absolute value, but not solving absolute value equations? How many will not even notice it went missing from the standards? Will anyone be willing to kill the little darling?

For years we've lamented math curricula in the United States that was "a mile wide and an inch deep." Given a fixed amount of time, going deeper means going narrower, and there is evidence that the CCSSM supports that (Porter, McMaken, Hwang, & Yang, 2011). For reasons I perceive to be mostly political, the Common Core State Standards don't go out of their way to specifically address content omitted compared to previous standards documents. In 1989, NCTM made a list of areas of emphasis and de-emphasis and many an argument in the math wars was waged using false dichotomies rooted in those lists. I'm sure those who wrote the CCSSM didn't want a repeat of those arguments, so now math teachers and those who support them are left to dig through the standards and find these omissions on their own.

(By the way, if you're looking for an interesting lesson for understanding and graphing absolute value, NCTM is offering one from a recent Mathematics Teacher for complimentary download: Angela Wade's "Teaching Absolute Value Meaningfully.")

References

Porter, A. C., McMaken, J., Hwang, J., & Yang, R. (2011). Common Core standards: The new U.S. intended curriculum. Educational Researcher, 40(3), 103–116. doi:10.3102/0013189X11405038

RYSK: Greeno, Pearson, & Schoenfeld's Implications for NAEP of Research on Learning and Cognition (1996)

This is the 14th in a series describing "Research You Should Know" (RYSK).

You might have read my recent post about Lorrie Shepard's 2000 article The Role of Assessment in a Learning Culture and assumed she focused on classroom assessment because changing large-scale, standardized assessments was a lost cause. Think again. By that time, an effort to integrate new theories of learning and cognition into the NAEP was already underway, traceable back to a 1996 report titled Implications for NAEP of Research on Learning and Cognition written by by James G. Greeno, P. David Pearson, and Alan H. Schoenfeld. For years Greeno has been recognized as one of education's foremost learning theorists, while Pearson and Schoenfeld are highly-regarded experts in language arts and mathematics education, respectively.

The National Assessment of Educational Progress, sometimes called "The Nation's Report Card," has been given to students in various forms since 1969. Unlike the high-stakes assessments given by states to all students, the NAEP is given to samples of 4th, 8th, and 12th grade students from around the country, and the use of matrix sampling means no student ever takes the entire test. The goal of the NAEP is to inform educators and policymakers about performance and trends, and details about how different NAEP exams try to achieve this are described in depth at the NAEP website.

Greeno et al. tried to answer two main questions in their report: (a) Does the NAEP inform the nation "about significant aspects of the knowing and learning" (p. 2) in math and reading, and (b) What changes in NAEP would make it a better tool for informing the nation about the performance and progress of our educational system? The authors acknowledge the tradition with what they call differential and behaviorist perspectives on learning, and focus more of their attention on the ability to assess cogntiive and situative perspectives, which have strong theoretical foundations but hadn't been reflected in most large-scale assessments.

Concisely, the report says the "key features of learning in the cognitive perspective are meaningful, conceptual understanding and strategic thinking" and that the "key feature of learning in the situative perspective is engaged participation with agency" (p. 3, emphasis in original). Greeno et al. say that if students are engaged in learning activities that reflect these perspectives, then the NAEP should try to capture the effects of those experiences.

One of the main reasons I'm writing about this report is because it gives me another chance to describe current learning perspectives that go beyond the simpler "behaviorism vs. constructivism" argument I knew as a teacher and heard from others. This report does this well without burdening the reader with all the gory details that learning theorists grapple with as they try to push these theories even further. So here's my summary of their summaries of each perspective:

Differential

This perspective accepts the assumption that "Whatever exists, exists in some amount and can be measured" (p. 10). For knowledge, that "whatever" is referred to as a trait, and different people have traits in different amounts. Evidence of traits can be detected by tests, and the correlation of different tests supposedly measuring the same trait is an indication of our confidence in our ability to measure the trait. Because the person-to-person amount of a trait is assumed to be relative, it's statistically important to design tests where few people will answer all items correctly or incorrectly.

Behaviorist

Behaviorism assumes that "knowing is an organized collection of stimulus-response associations" (p. 11). To learn is to acquire skills (usually and best in small pieces) and measuring learning is seen as an analysis of behaviors which can be decomposed into responses to stimuli. Behaviorism's influence on curriculum is seen when behavioral objectives are organized as a sequence building bigger ideas out of smaller, prerequisite objectives.

Cognitive

The cognitive perspective primarily focuses "on structures of knowledge, including principles and concepts of subject-matter domains, information organized by schemata, and procedures and strategies for problem solving and reasoning" (p. 12). Learners actively construct their knowledge rather than accept it passively, and conceptual understanding is not just the sum total of facts. The early part of the cognitive revolution was reflected in the math and science reforms of the 1950s and 1960s, while Piagetian ideas and research on student understanding have pushed the perspective further. Assessments need to determine more than right and wrong answers, and research involving think-aloud protocols, student interviews, eye-tracking studies, and patterns of responses have yielded better theories about how to assess for student understanding.

Situative

The situative perspective is a social view of learning focused on "interactive processes in which people participate in practices that are organized by the societies and communities they belong to, using the technologies and natural resources in their environments" (p. 14). Knowing is no longer in the head -- instead it is seen as participation in a community, and learning is represented by increased and more effective participation. John Dewey took parts of this perspective in the early 20th century, but we owe much of the theory to Lev Vygotsky, whose work in the 20s and 30s in the Soviet Union eventually emerged and has heavily influenced learning science since the late 1970s. The situative perspective is more readily applied to interactions between people or between people and technology (which is seen as a cultural artifact with social roots), but even solitary learners can be assessed with the situative perspective if we focus on "the individual's participation in communities with practices, goals, and standards that make the individual's activity meaningful, either by the individual's adoption of or opposition to the community's perspective" (p. 14). The influence of the situative perspective on curriculum and classrooms is most easily seen in the focus on student participation, project work, small-group discussions, and authentic work in subject-area disciplines.

In summary, achievement in each perspective can be described as:
Differential/Behaviorist
- "progress a student has made in the accumulation of skills and knowledge" (p. 16)
Cogntive
- a combination of five aspects (pp. 16-18):
  1. Elementary skills, facts, and concepts
  2. Strategies and schemata
  3. Aspects of metacognition
  4. Beliefs
  5. Contextual factors
Situative
- a combination of five aspects (pp. 19-21):
  1. Basic aspects of participation
  2. Identity and membership in communities
  3. Formulating problems and goals and applying standards
  4. Constructing meaning
  5. Fluency with technical methods and representations

What Does This Mean for the NAEP?

Greeno et al. declared that the NAEP was "poorly aligned" (p. 23) with the cognitive perspective. It hadn't captured the complexity of student knowledge and they recommended a greater focus on problems set in meaningful contexts and tasks that reflected the kind of knowledge models and structures theorized in the research. As for the situative perspective, Greeno et al. went so far to say that what the NAEP had been measuring was "of relatively minor importance in almost all activities that are significant for students to learn" (p. 27). Whereas the situative perspective focuses on participation in a particular community or knowledge domain, it's impossible to escape the reality that on the NAEP, the domain is test-taking itself, a "special kind of situation that is abstracted from the variety of situations in which students need to know how to participate" (pp. 28-29). Measuring learning from the situative perspective would require a complicated set of inferences about a student's actual participation practices in an authentic domain, and the technical limitations of the NAEP limits our ability to make those inferences.

The report continues with specific details about how we might measure learning in language arts and mathematics with the NAEP from both a cognitive and situative perspective. In the conclusion, the authors first recommend some systemic changes: First, NAEP needed more capacity for attending to the long-term continuity of the test and its design. Given how important NAEP is for measuring longitudinal trends, we can't change it without a careful study of how to compare new results to old. Second, the authors wanted a national system for evaluating changes in the educational system. The NAEP alone can't tell us everything we need to know about the effectiveness of educational reforms.

As for recommendations for the test itself, Greeno et al. emphasized the need to align the assessment with ongoing research, especially in the cognitive perspective. Instead of planning for NAEP tests one at a time and contracting out various work, the development process needed to become more continuous with particular sustained attention given to progress in the cognitive and situative dimensions. More ambitiously, the authors recommended a parallel line of test development to begin establishing new forms of assessment that might capture learning in these newer perspectives. This is a critical challenge because while we know the least about assessing from the situative perspective, the situative is often the perspective that frames our national educational goals. The NAEP can't measure progress to situative-sounding goals without better measurement of learning from a situative perspective.

It has now been 12 years since the release of this report. I don't know how Greeno et al.'s recommendations have specifically been followed, but there is good news. If you read most any of the current NAEP assessment frameworks, you can find evidence of progress. The frameworks have changed to better measure student learning, particularly from the cognitive perspective. Some frameworks honesty address the difficulty in measuring the situative perspective using an on-demand, individualized, pencil-and-paper (but increasingly computer-based) test. (See Chapter One of the science framework, for example.) Will we see any radical changes any time soon? I doubt it. The information we get about long-term trends from the NAEP requires a certain amount of stability. Given the onset of new national consortia tests based on the Common Core State Standards, I think the educational system will get its fill of radical change in the next 3-5 years. With that as the comparison, we all might contently appreciate the stability and attention to careful progress reflected in the NAEP.

References

Greeno, J. G., Pearson, P. D., & Schoenfeld, A. H. (1996). Implications for NAEP of research on learning and cognition (p. 84). Menlo Park, CA.

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

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

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

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

History

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

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

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

Modern Theories

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

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

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

Improving Assessment

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

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

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

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

Dynamic Assessment

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

Prior Knowledge

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

Feedback

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

Transfer

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

Explicit Criteria

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

Self-Assessment

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

Evaluation of Teaching

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

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

Conclusion

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

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


References

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

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

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

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

Bonnie H. Litwiller, 1937-2012

I got word tonight that my undergraduate advisor, Bonnie Litwiller, passed away a couple days ago at the age of 74.

As a freshman at UNI, I had a temporary advisor until my program became more certain. After declaring as a math education major, Bonnie Litwiller was assigned as my advisor. I knew nothing about her. I remember asking Ed Rathmell, whom I had gotten to know while applying for a scholarship, what to expect from Litwiller as an advisor. I remember his response: "If you listen to her and do what she asks, she's great. She'll have your back when you need something. But don't cross her."

That's an uneasy way to know someone before you even get a chance to meet them. It felt like a description more fitting of mafia boss than a professor. But Rathmell's advice, as usual, was solid. Litwiller proved to be tough, and she made it clear to us that being a good math teacher was hard work. She set a good example: she and her research partner, David Duncan, would set aside a day a week where they'd lock themselves away in the library and write. As UNI isn't a top-level research university, the research activities of professors aren't always visible to the students. But Litwiller's dedication to research was clear, and there was no topic too small or journal too obscure. If she thought she had knowledge that would improve the teaching and learning of mathematics somewhere -- anywhere! -- she would write and submit for publication. She continued to write and publish even after her retirement from UNI in 2000, eventually passing the almost unfathomable mark of 1000 scholarly publications.

I took two classes with Litwiller, Teaching Middle School Mathematics and Teaching High School Mathematics. The classes were tough due to Litwiller being both picky about the quality of our work and her lack of clarity in describing what she wanted us to do. Some of us thought she was just being careless with her assignments, but I always wondered if this wasn't somehow purposeful. Either way, it was clear that she didn't want to do a lot of hand-holding. Some of us, ever so quietly yet respectfully, referred to her as the bulldog. A trusty companion that might just bite if you got out of line. If you didn't have the initiative and sense of responsibility to do quality work, I think she wanted a way for you to sort yourself out of the program. It happened, too; every semester some classmate would go missing and we'd try to find out what happened. Inevitably, someone would say, "They couldn't cut it. Litwiller dropped them from the program." You hear a lot today about colleges adopting GPA or test score requirements to improve the quality of their education majors. We didn't have those -- we had Litwiller. And just like letting a GPA decide who can be a teacher, I'm sure her judgement wasn't perfect and mistakes were made. (An acquaintance of mine, who shall remain nameless but now holds a PhD in math education, told me about a narrow escape from Litwiller's axe after a dispute over access to a local school.) But I think Litwiller had a sense for the quality that people expected from a UNI-prepared teacher, and a sense for giving us some survival skills that would get us through our first few years of teaching. There must have been far more successes than failures, too -- by the time I graduated in 1999, someone had estimated that a quarter of all the math teachers in the State of Iowa had been taught by Bonnie Litwiller.

My appreciation for Litwiller and her work has grown through my years first as a teacher and now as a graduate student in mathematics education. It was she who first introduced me to the NCTM Standards, and her direction of the NCTM Addenda Series was and still is an enormous contribution to the field of math education. What I believe was originally intended to be a six-book series to support the Standards grew into 22 total books, each designed to take the research behind the Standards and turn it into something teachers could use. Litwiller might have been the director and not the author of the Addenda Series, but it carried her trademark: getting as much useful information into the hands of teachers as possible. She gave me two books from the series, the middle school and high school books about statistics and data analysis. It was the first time I really thought about statistics education, and it's since become the area of school mathematics I find most interesting.

The world will miss Bonnie Litwiller, but she didn't leave without making a mark, both on the field of mathematics education and on me. Teacher education is a challenging business, and it's probably best to judge it with a certain amount of hindsight. For all of her toughness, she did have my back when I needed it, just as Ed Rathmell said she would. I may not have learned all that she tried to teach me, but maybe her most important lessons -- a sense of dedication and rejection of "good enough" -- have been most helpful in getting me to where I am today.

Sorting Out the Summative: When Standards-Based Grading Meets the End of the Semester

Source: Wikipedia

Many teachers who choose to use standards-based grading eventually find themselves facing the reality of their school's grading policies and tradition: the expectation of final, summative grades that are reported as percentages and letters. So regardless how hard you try to focus on quality feedback instead of grades all semester long (for good reason), there comes a time when, for reasons probably beyond your control, you have to turn levels and descriptions of student understanding into numbers. This is SBG's "Monday Morning Problem" that doesn't always get addressed in theory. But this week is finals week for my basic statistics students, so for me the time has come to convert standards-based formative grades into a summative grade, including calculating final exam grades. Here I'll try to describe the two steps I'll take to calculate my students' grades: (a) conversion of their formative scores into a summative score and (b) scoring and inclusion of the final exam into their semester grades.

Formative to Summative
Besides giving students a lot of written and verbal feedback about where they should try to improve, I've been using the simplest of measures to record their performance on class objectives: either students (a) "get it," (b) "sort of get it," or (c) "dont' get it/haven't demonstrated it." You could think of these as "green light," "yellow light," and "red light," respectively. I've tried discerning more levels of understanding in a gradebook and it only seems to lead to confusion and indecision (both for me and students), so I'm sticking to three levels, as suggested in Her & Webb (2004). If I need more detail, I can always go back to the copies of the work students have submitted and the comments I've made.

The gradebook we have for class is pretty primitive and as far as I can tell it only accepts numbers, so I mark my three levels as either a 2, a 1, or a 0. It doesn't take much explaining to students that a 1 shouldn't be viewed as "out of two" and therefore worth 50%. I do tell them, though, that in order to receive credit for the course they should average a 1 across all objectives. In other words, you can't pass the class without an average of at least some understanding of every objective.

Around here and in many other places, 70% seems to be the low end of passing grades. (We're not messing with Ds.) So if a student with all 1s should get at least a 70%, and a student with all 2s maxes out at 100, and we choose a linear function between the two, the "conversion formula" to percentages is simply:

percentage = 30 * objective score average + 40

If you feel a little dirty at this point because you know you just reduced all the various skills, knowledge, and abilities of your students into a single number, I say join the club. If you didn't feel that way I wouldn't have expected you to be using standards-based grading to begin with.

A "No Surprises" Approach to Final Exam Grades
Designing a final exam is often tricky business. It can't possibly assess everything in the course, but we generally want it to include the major topics and themes for the class and be possible to complete in the time allowed. We also have to think about difficulty. Trust me, your students are!

Teachers want their finals to be challenging, but they don't want to have that sinking feeling as they grade the exams that maybe the test was too hard. For whatever reason, sometimes students perform poorly and averaging the final exam grade into their other grades will look like a disaster. But ask yourself: What am I more confident in, my careful judgments of students' ability as demonstrated over an entire semester, or a fleeting, one-time judgement of students' ability on a single assessment during the most stressful time of the year? If you're using standards-based grading, I already know how you'll answer that question. If not, consider this example: I have a student who I know can do stats. She's turned in good work. She's asked quality questions. We've had good discussions. But I also know she has seven final exams this week. I still think she'll do fine, but I'll understand if she's not at her best. And I need a grading system that reflects that understanding.

In order to free myself to still give challenging, yet reasonable, assessments, without risking any huge surprises when grades are calculated, I perform a little statistical magic that ensures that the distribution of final grades has the same center and spread of class grades before the final. I'm sure many of you try "curving" your exam scores some other way, such as letting the top score count as the total possible, or even having a pre-set distribution in mind of how many As, Bs, Cs, etc. you'll allow (which is not a good idea, generally, for reasons described by Krumboltz & Yeh, 1996). I prefer my method because it accounts for the distribution of grades, not just the top score, and the distribution is determined by the students, not arbitrarily by me. Allow me to demonstrate with a couple examples.

Suppose before the final the average percentage grade is 85 and the standard deviation of those grades is 10. Then I grade my final exams and find that the average final exam grade is 60 with a standard deviation of 18. Ouch. But don't worry -- statistics will come to our rescue.

Provided you know a little basic descriptive statistics, the conversion is simple. For each student's final exam score, find out how many standard deviations above or below the mean they scored on the final (their final exam z-score), and match that with the same number of standard deviations above or below the mean they'd fall on the pre-final grade distribution (their pre-final z-score). Consider the following students and the class and exam statistics above:

  • Suppose Student A scores a 51 on the final exam. That's 0.5 standard deviations below the mean. (51 - 60 = -9, and -9/18 = -0.5.) So where is 0.5 standard deviations below the mean on the pre-final distribution? If that mean is 85 and the SD is 10, then 0.5 standard deviations below the mean is 80. So I record an 80 for that student instead of a 51.
  • Suppose Student B scores a 75 on the final exam. That's about 0.83 standard deviations above the mean. (75 - 60 = 15, and 15/18 = 0.83.) So where is 0.83 standard deviations above the mean on the pre-final distribution? About 8.3% above an 85, so I record their exam grade as a 93.3.
  • Suppose Student C scores a 60 on the final exam. That's the same as the mean, so zero standard deviations above or below. That conversion is super-easy: their final exam grade is the mean of the pre-final mean, an 85.
For an example of how to set up a spreadsheet to do this, see https://docs.google.com/spreadsheet/ccc?key=0Anne5Z-jCkqhdDVtemkyaGhnRWFfclJoa0dIUVQ5RVE. I recommend making a copy of it for yourself and seeing what happens as you change values.

This is not a perfect system (and comments about its imperfections are welcome in the comments), but it does take away the element of surprise if the final exam happens to be way too easy or too difficult, or if other circumstances prevent grades from working out the way you'd expect. Yes, this is a norm-referenced system instead of a criterion-referenced system, meaning that the grades students earn on the final is measured largely as how they compare to their classmates and the class average. The good news is this: both the teacher and the students have an incentive before the final to master as many objectives as possible, and that is criterion-referenced. A high pre-final average helps everyone get a high final exam average, and a small pre-final standard deviation minimizes variability in final exam scores.

References

Her, T., & Webb, D. C. (2004). Retracing a path to assessing for understanding. In T. A. Romberg (Ed.), Standards-based mathematics assessment in middle school: Rethinking classroom practice (pp. 200-220). New York, NY: Teachers College Press.

Krumboltz, J. D., & Yeh, C. J. (1996). Competitive grading sabotages good teaching. Phi Delta Kappan, 78(4), 324-326. Retrieved from http://www.jstor.org/stable/20405782