I don’t think the normal notifications went out about this, so I’m adding this to let people know about the collection of essays about secondary mathematics that I posted yesterday.
In 2008–2009 Dick Stanley and Phil Daro, with the help of Vinci Daro and Carmen Petrick, convened a group of mathematicians and educators to write essays clarifying the mathematical underpinnings of secondary school mathematics in the United States. At the urging of Dick Stanley I am publishing these essays here.
Some people are getting a message that comments have been closed if they try to post a comment on a post. I don’t generally close comments, so this is an error. I haven’t figured out how to fix it, but if this happens to you please send me an email.
A number of people have gotten in touch with me recently about Appendix A, so I wanted to clarify something about its role. States who adopted the standards did not thereby adopt Appendix A. The high school standards were intentionally not arranged into courses in order to allow flexibility in designing high school courses, and many states and curriculum writers have taken advantage of that flexibility. There was a thread about this on my blog 3 years ago, and there is a forum on the topic here.
Appendix A was provided as a proof of concept, showing one possible way of arranging the high school standards into courses. Indeed, on page 2 of the appendix it says:
The pathways and courses are models, not mandates. They illustrate possible approaches to organizing the content of the CCSS into coherent and rigorous courses that lead to college and career readiness. States and districts are not expected to adopt these courses as is; rather, they are encouraged to use these pathways and courses as a starting point for developing their own.
States will of course be constrained by their assessments. But Smarter Balanced consortium does not have end of course assessments in high school, leaving states and districts free to arrange high school as they choose. And although PARCC does have end of course assessments, they do not follow Appendix A exactly. See the footnote on page 39 of the PARCC Model Content Framework , which says
Note that the courses outlined in the Model Content Frameworks were informed by, but are not identical to, previous drafts of this document and Appendix A of the Common Core State Standards.
Furthermore, there are plenty of states not using either the PARCC of SMARTER Balanced assessments.
I hope this helps clear things up.
Here the almost final draft of the Progression on Number and Operations in Base Ten, K–5. It incorporates many changes in response to comments here on this blog and elsewhere.
In addition to numerous small edits and corrections, and some redrawn figures, here are some of the more significant changes:
- The sidenote with glossary entry for algorithm was moved to first instance of “algorithm” together with some text on notation for standard algorithm (this piece is a revision of a paragraph that was in the main body of the previous version).
- Section on Strategies and Algorithm: The 2 old paragraphs were deleted and 3 new paragraphs were inserted. Reason: the new paragraphs give an overview of the organization of the NBT standards for strategies and algorithms explaining that students see efficient, accurate, and generalizable methods from the beginning of their work with calculation and that there is a progression from strategies to algorithms: for addition and subtraction (with whole numbers in K to Grade 4; and generalization to decimals in Grades 4 to 6), for multiplication (Grades 3 to 5) and division (Grades 3 to 6) with whole numbers, then decimals.
- The balance of emphasis on “special strategy” vs “general method” in the earlier progression has been shifted in this draft in the direction of general methods..
- Mathematical practices section was revised to focus more on the centrality of the SMPs, illustrating progression from strategy to algorithm and following the structure of the sections on computations, and strategy and algorithm.
As usual, please comment in NBT thread in the Forums.
Illustrative Mathematics has a new look today. There’s a video explaining some of the new features on the Illustrative Mathematics Facebook page. One big new feature is the course blueprints. At the moment we just have K–5 blueprints. We’ll be adding more content to those and also adding high school and middle school blueprints over the next few months. I’ve made a forum here for people to comment and ask questions about them.
Standards shouldn’t dictate curriculum or pedagogy. But there has been some criticism recently that the implementation of CCSS may be effectively forcing a particular pedagogy on teachers. Even if that isn’t happening, one can still be concerned if everybody’s pedagogical interpretation of the standards turns out to be exactly the same. Fortunately, one can already see different approaches in various post-CCSS curricular efforts. And looking to the future, the revisions I’m aware of that are underway to existing programs aren’t likely to erase those programs’ mutual pedagogical differences either.
Of course, standards do have to have meaningful implications for curriculum, or else they aren’t standards at all. The Instructional Materials Evaluation Tool (IMET) is a rubric that helps educators judge high-level alignment of comprehensive instructional materials to the standards. Some states and districts have used the IMET to inform their curriculum evaluations, and it would help if more states and districts did the same.
The criticism that I referred to earlier comes from math educator Barry Garelick, who has written a series of blog posts that aims to sketch a picture of good, traditional pedagogy consistent with the Common Core. The concrete proposals in his series are a welcome addition to the conversation math educators are having about implementing the standards. Reading these posts led me to consider the following question:
If the only computation algorithm we teach is the standard algorithm, then can we still say we are following the standards?
Provided the standards as a whole are being met, I would say that the answer to this question is yes. The basic reason for this is that the standard algorithm is “based on place value [and] properties of operations.” That means it qualifies. In short, the Common Core requires the standard algorithm; additional algorithms aren’t named, and they aren’t required.
Additional mathematics, however, is required. Consistent with high performing countries, the elementary-grades standards also require algebraic thinking, including an understanding of the properties of operations, and some use of this understanding of mathematics to make sense of problems and do mental mathematics.
The section of the standards that has generated the most public discussion is probably the progression leading to fluency with the standard algorithms for addition and subtraction. So in a little more detail (but still highly simplified!), the accompanying table sketches a picture of how one might envision a progression in the early grades with the property that the only algorithm being taught is the standard algorithm.
The approach sketched in the table is something I could imagine trying if I were left to myself as an elementary teacher. There are certainly those who would do it differently! But the ability to teach differently under the standards is exactly my point today. I drew this sketch to indicate one possible picture that is consistent with the standards—not to argue against other pictures that are also consistent with the standards.
Whatever one thinks of the details in the table, I would think that if the culminating standard in grade 4 is realistically to be met, then one likely wants to introduce the standard algorithm pretty early in the addition and subtraction progression.
Writing about algorithms is very difficult. I ask for the reader’s patience, not only because passions run high on this subject, but also because the topic itself is bedeviled with subtleties and apparent contradictions. For example, consider that even the teaching of a mechanical algorithm still has to look “conceptual” at times—or else it isn’t actually teaching. Even the traditional textbook that Garelick points to as a model attends to concepts briefly, after introducing the algorithm itself:
This screenshot of a Fifties-era textbook is as old-school as it gets, yet somebody on the Internet could probably turn it into a viral Common-Core scare if they wanted to. What I would conclude from this example is that it might prove difficult for the average person even to decide how many algorithms are being presented in a given textbook.
Standards can’t settle every disagreement—nor should they. As this discussion of just a single slice of the math curriculum illustrates, teachers and curriculum authors following the standards still may, and still must, make an enormous range of decisions.
This isn’t to say that the standards are consistent with every conceivable pedagogy. It is likely that some pedagogies just don’t do the job we need them to do. The conflict of such outliers with CCSS isn’t best revealed by close-reading any individual standard; it arises instead from the more general fact that CCSS sets an expectation of a college- and career-ready level of achievement. At one extreme, this challenges pedagogies that neglect the key math concepts that are essential foundations for algebra and higher mathematics. On the other hand, routinely delaying skill development until a fully mature understanding of concepts develops is also a problem, because it slows the pace of learning below the level that the college- and career-ready endpoint imposes on even the elementary years. Sometimes these two extremes are described using the labels of political ideology, but I have declined to use these shorthand labels. That’s because I believe that achievement, not ideology, ought to decide questions of pedagogy in mathematics.
Jason Zimba was a member of the writing team for the Common Core State Standards for Mathematics and is a Founding Partner of Student Achievement Partners, a nonprofit organization.
The following lectures are scheduled in the series on Thursday nights from 7-8pm Eastern on Adobe Connect. Watch them live with the ability to ask questions, or watch the recordings at any time:
September 25, 2014 Linda Gojak, Immediate Past President, National Council of Teachers of Mathematics, Director, The Center for Mathematics Education, Teaching, and Technology, John Carroll University “Using Representations to Introduce Early Number and Fraction Concepts”
October 23, 2014 Dona Apple, Mathematics Learning Community Project, Regional Science Resource Center, University of Massachusetts Medical School “Supporting students’ conceptual understanding about number through reasoning, explaining and evidence in both their oral and written work”
November 20, 2014 Brad Findell, The Ohio State University
December 11, 2014 Francis (Skip) Fennell, Professor of Education McDaniel College, Past President NCTM “Fractions Sense – It’s all about understanding fractions as numbers (and this includes those special fractions – decimals!) – use of representations, equivalence, comparing/ordering and connections”
January 22, 2015 Susan Jo Russell, TERC: Mathematics and Science Education and Deborah Schifter, Education Development Center (EDC) “Operations and Algebraic Thinking in the Elementary Grades”
This school year we will offer two series. In the fall we are featuring “Working with Number in the Elementary Classroom” and this spring we will offer “Incorporating Mathematical Practices into the Middle and High School Classroom.” The intended audience for these series is classroom teachers, district and state mathematics specialists, and mathematics coaches. The five hour long sessions will include 40 minutes of presentation from national experts on Adobe Connect, followed by 20 minutes of Q&A. The sessions will also be recorded for participants that are not able to join in person. The cost to virtually attend each series is $150.
Here is a flyer to circulate among friends that might be interested or to post in the staff room! Hope to see you there.
About a year ago I noticed there was a lot of misinformation being spread about the process for writing the standards, so I came up with the brilliant idea of pointing people to the historical source documents that chronicled the process: the NGA press releases about the Common Core during 2009–2010. That will solve the problem, I thought; people will just read the press releases and figure it out. Boy was I ever wrong. In this post I’ll try to give a clearer timeline of the process. Along the way I’ll point out the involvement of testing organizations, since I think that one of the reasons the misinformation has survived for so long is a narrative, compelling to some, that the testing industry dominated the process. (Spoiler alert: they didn’t.)
First, here is the list, with an additional one from July 2009 that I missed last time (which has been the source of much confusion):
- 1 June 2009, Forty-Nine States and Territories Join Common Core Standards Initiative
- 1 July 2009, Common Core State Standards Development Work Group and Feedback Group Announced
- 1 September 2009, Fifty-One States And Territories Join Common Core State Standards Initiative
- 21 September 2009, Common Core State Standards Available for Comment
- 24 September 2009, Common Core State Standards Initiative Validation Committee Announced
- 10 November 2009, Common Core State Standards K-12 Work and Feedback Groups Announced
- 10 March 2010, Draft K-12 Common Core State Standards Available for Comment
- 2 June 2010, National Governors Association and State Education Chiefs Launch Common State Academic Standards
Notice that there seem to be duplicate announcements of the Work and Feedback Group and duplicate releases of the standards. What’s going on here is that there were two documents. First, in summer of 2009, the people listed in the July 2009 release worked on the document that was announced in September of 2009. That document, which was actually entitled College and Career Readiness Standards for Mathematics, was confusingly referred to as Common Core State Standards in the title of the September 2009 press release. If you take a look at it you will see that it is a draft description of what students should know by the end of high school.
Subsequently, as described in the November 2009 press release, a new process with new groups was started, to produce “K–12 standards.” These were to be a set of grade level recommendations that described a pathway to college and career readiness. For the K–12 process, there were about 50 people on the Work Team and about 20 people on the Feedback Group for mathematics, representing a wide range of professions, including teachers, mathematicians, policy makers, and one representative each from College Board and ACT … none representing for-profit providers of assessments. The members of this group are listed in a linked pdf in the press release. This is the document that was released for public comment in March 2010, as described in the March 2010 press release, and released in final form as the Common Core State Standards on 6 June 2010, as described in the final press release.
As you can see from the list of members, I chaired the Work Team for the second document. Within the work team there was a smaller writing team consisting of myself, Jason Zimba, and Phil Daro (who had all been involved in the summer 2009 document, Phil Daro as chair for mathematics). We based the standards on narrative progressions of particular mathematical topics across grade levels that were solicited from the Work Team. We circulated many drafts to the Work Team, the Feedback Group, the 48 participating states, various national organizations such as AFT and NCTM, and, in March 2010, the public (see the March 2010 release). I personally made sure that we responded to and made considered decisions about all of the voluminous feedback we received.
When you hear people claim that “the standards were written by the testing industry,” they are probably referring to the first document, because of the greater involvement of College Board and ACT. Both organizations, along with Achieve, which was also represented, had conducted research into the requirements of college and career readiness. (All are non-profits, by the way.) The problem is that some people refer to the first document in a way that suggests they are talking about the second document (i.e., the actual K-12 standards adopted by states). That is an error and a misleading one.
The two documents are different in nature, of course, since one of them is just a picture of an endpoint while the other is a progression. Feel free to compare them. One influence of the first document on the second is that in the first document you can see the first draft of what became the Standards for Mathematical Practice. And the topic areas listed in the first document evolved into the high school conceptual categories in the second. All this evolution happened under the processes for the second document, with input from the various groups described above.
I think the second document is the work of the 70-odd people listed as the Work Team and Feedback Group in the November 2009 press release. But, just for fun, I put the teams for the two documents together and counted how many of them came from ACT and College Board (no other testing organizations were represented). It comes to a total of 81 people with 7 from ACT and College Board, about 9%. So even with this interpretation the claim that the process was dominated by the testing industry is false.