Researching the Common Core
I’m in the middle of researching the Common Core standards for math. So far I’ve watched a Diane Ravitch talk, which I blogged about here, which was interesting but raised more questions than it answered, at least for me.
I’ve also interviewed Bill McCallum this week, who was a lead writer and chair of the Work Team that wrote the Common Core standards for mathematics. I’m still writing up that interview but I should have it done soon.
Next up I plan to interview a long-time teacher and current principal of a Brooklyn-based girls school for math and science, Kiri Soares, on her perspective on the Common Core standards and standardized tests in general.
One thing I can say already for sure: people who are not insiders here conflate a bunch of different issues. I’m hoping to at least separate them and understand where people stand on each issue, and if I at the very least get to the point of agreeing to disagree on well-defined points then I will have done my job.
Tell me if you think I need to go further to fully understand the issues at hand. Of course one thing I’m not doing is delving directly into the content of the standards, and that may very well be essential to understanding them. I’d love your thoughts.
mathbabe 
Talk to Jose Vilson (The Jose Vilson Project). He’s located in New York and he can help you break out some math-related issues–middle school math teacher, very visible, has a book coming out soon. Additionally, Mercedes Schneider (deutsch29) has done a LOT of research into the Common Core and her website (http://deutsch29.wordpress.com/) is an incredible resource on all things Common Core. Also, follow what Common Core is doing (or not doing) for higher level special education students.
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Seems to me I’ve heard positive reviews from many high school math teachers and negative from everyone else. Since you’re focusing on math, the issues with language arts don’t matter. But please check into age appropriateness for young students. Do the math standards contribute to the loss of playtime in kindergarten?
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I think the question of play (in kindergarten, in primary school, in secondary school, in life) is driven much more by other factors than the particular curriculum. There are two dimensions:
– does the school/teacher/parent believe in play?
– does the school/teacher/parent know the subject well enough to have confidence to encourage play?
The common core standards in math are basically orthogonal to those two questions. However, for someone who believes in play, but doesn’t really know how to implement it, the core standards can be helpful in the following procedure:
– read the standards
– play a bunch of games and think about other games
– recognize where the concepts from the standards appear naturally in the games
– proceed to have your kids play those games secure in the knowledge that you can now use the common core standards as justification for any (non-player) doubters among the school administration and parents.
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And it may have been conflated by many, but good standards are not useful if the tests don’t really reflect the good parts. The standards and the tests seem to be pretty tightly linked.
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Two things: first, why not interview David Coleman? As you’re undoubtedly aware, he led the development of the Common Core standards and is now head of ETS. Diane Ravitch has written ambivalently and wonderingly about him, questioning his commitment or valuation of literature since only 25% of Common Core content in secondary school is devoted to it. While her question re “why only 25%” is valid, this doesn’t mean Coleman is a philistine. This conversation would give you the opportunity to ask the critical questions and concerns you have about the extent to which the Common Core is beholden to corporate interests.
Second, testing and evaluation of the standards seems to have taken a back seat during development but will be absolutely key in their implementation and success. How “testable” are some of the standards and what are the options?
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Good idea to interview him. But David did not lead the development of the Common Core. He was one of the three lead writers for the English Language Arts standards, along with Sue Pimentel and Jim Patterson. Sue also chaired the larger work team for ELA, as I did for Math. The two teams (Math and ELA) were jointly managed by CCSSO and NGA. On the CCSSO side the point person was Chris Minnich, who was CCSSO’s director of standards, assessment, and accountability at the time. Chris is now Executive Director of CCSSO. On the NGA side it was Dane Linn, who was director of education for NGA’s Center for Best Practices. Day to day management of the work was largely in Chris’s hands; he is the one who handled sending the standards out to states for review and collecting feedback, taking calls from individuals with comments, and so on.
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The curriculum itself does seem like something you should look at, or at least see how other experts have evaluated it. Wasn’t one of the problems with the development of the core that no one with early childhood expertise was consulted? I’d be interested in what the early childhood folks are saying, due to the ways in which older core standards are having a recursive effect on K-2.
I’d talk to teachers who aren’t currently in administration; look over what NYSUT and other union bodies have said in resolutions and other union materials, possibly interviewing some folks there; also perhaps parents who are educators who are thinking about the testing from their kids perspectives–there’s a group Parents against Testing, I think, that includes teachers.
It seems like there are many layers here, about development, implementation, content and evaluation. Most teachers I’ve spoken with seem to have fewer problems with specific content in particular subject areas and more with implementation, support, and the tests. On that last, the lockstep approach of yearly performance standards will likely force standardized teaching as well, which will hurt many students and many teachers. But, an overarching problem I hear about is the extent to which the particular standards crowd out everything beyond reading and math. So in that sense, I think to evaluate math standards you’d need to place them in the larger context of the whole curriculum, and what you can and should expect from students–particularly young ones, who are getting tested younger as a result of the CC.
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I think you are using “curriculum” incorrectly here. The Common Core is a standard for what topics (and subtopics) should be taught. A curriculum is the materials, processes, etc. which are intended to teach students the particular topics in question. While what is in the standard will certainly affect the curriculum (and might even force to some extent
particular teaching methods to be used), they are not the same thing. There are any number of textbooks available (or coming out) which purport to cover CC, each of them would be part of distinctive curriculums. Likewise, the CC is not any particular test. Nor does using CC to guide curriculum development inherently require a testing regimen of any sort. Unfortunately advocates of testing and “teacher responsibility” have managed to conflate the CC standards with many other things to the extent that both proponents and opponents of those other things don’t even seem to realize that they aren’t the same thing.
Your point about CC standards potentially crowding out other topics may be true, but that is going to inherently occur whenever a decision is made to cover one topic rather than another. Locally, whether to do foreign language instruction in K-5 is a topic of discussion.
Nobody is inherently against the idea, but the advocates for doing so never talk about what other topic is going to have less time spent on it. In that sense, going to CC for a topic might mean less time for other non-CC subjects due to increased topics to be covered.
Or maybe not, if it is just a different set of topics. I suspect though that it is the
testing regimens rather than any particular CC standard which is going to cause the most loss of time for non-CC subjects. And it should be noted that testing regimens have been on the rise for some time and preceded CC.
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On[e] thing I can say already for sure: people who are not insiders here conflate a bunch of different issues.
The word “conflate” suggests (to me) the attitude that the issues have been wrongly combined. While perhaps true in some cases, keep in mind that these different issues may have been conflated by design. Or, at least, practically conflated through policy.
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Cathy,
I look forward to your analysis of the Common Core.
There are a number of issues that come into play, in my view.
First of all, there is the unusual way that these standards were drafted — in secret, with little participation by K12 educators. They were then “adopted” in many states with little input from educators or the public. This was brought about by coordinated actions of the Gates Foundation and the Department of Education. Some defenders of the Common Core wave this off, as if all that matters is the content of the standards. But the content of the standards reflects its origins and intent. And I think it sets a dangerous precedent to allow a corporate philanthropy to essentially buy the process by which national education standards are set.
The second huge issue is the fact that these standards are deeply embedded in a standardized testing accountability paradigm. They were written to be tested, and the driving assumption is that all students must be “on grade level” starting in Kindergarten, to ensure that they will emerge from high school “career and college ready.” Standards have been backward mapped from that assumption, leading to some topics being pushed down to grade levels where they are inappropriate. Early childhood educators have been especially concerned about this.
We have seen the impact of the new, “more rigorous” Common Core tests in New York, where they were given last spring. Only about 30% of the students passed. The numbers were much worse for African American students, and English learners and Special Ed were worst of all. When these results are used for promoting and retaining students, and teacher evaluation, and deciding to close schools, the results will be devastating.
Here is what I wrote a few months ago: http://blogs.edweek.org/teachers/living-in-dialogue/2013/11/common_core_standards_ten_colo.html
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The standards were not drafted in secret; there was a huge amount of participation by K-12 educators; states who adopted had multiple rounds of review before adopting. The Gates Foundation’s provided funding, and, later, after many states had adopted (maybe 20 or so), the Department of Education did provide incentives for adoption. I know because I was there. But you don’t have to believe me, just do some basic fact checking. For example, go to NGA’s website and search their press releases for Common Core between 2009 and 2010. Or, if you can’t be bothered doing that, I’ve collected them at http://commoncoretools.me/2013/06/10/learning-about-the-standards-writing-process-from-nga-news-releases/.
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Bill,
I am not sure if you got all the pertinent releases. I wrote a post you may have read before, on July 6, 2009, which was in response to the release from the NGA describing the process. That release stated the following: “The Work Group’s deliberations will be confidential throughout the process.”
That means secret. This was even celebrated in media reports:
“the San Francisco Chronicle, wrote last month [June, 2009] that secrecy in this project is “… a wise decision. A truly open process would result in the experts being lobbied by countless interest groups, and – given the still-controversial nature of national standards – it could torpedo the plan altogether.”
There is a distinction between the initial drafting of a document, and the feedback process. There is no evidence that K12 educators were involved in the drafting of the standards, although there were some who were allowed to give feedback after the draft standards were written.
http://blogs.edweek.org/teachers/living-in-dialogue/2009/07/national_standards_process_ign.html
And there are different views among the teachers who participated in the review process. Not all felt their concerns were heard. Here is one point of view you will not find on the NGA web site: http://blogs.edweek.org/teachers/living-in-dialogue/2013/11/florida_teacher_i_was_among_th.html
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Hi Anthony, I already commented on your use of that press release on Cathy’s blog, here: https://mathbabe.org/2014/02/11/interview-with-bill-mccallum-lead-writer-of-math-common-core/#comment-59820. Also on your assertion that there is no evidence that K–12 educators were involved in the drafting. I gave a link to the list of members of the work team, which includes K–12 educators. I’m not talking about the feedback group here, that was a different group. I managed the writing process, and I can assure you the work team was involved; also teams of teachers assembled by AFT and by many of the participating states. We were flooded with feedback from teachers, and responded to it. One of those teachers was Becky Pittard, who was both on the Work Team and on the AFT team. She is quoted directly refuting your assertion here: http://www.achieve.org/more-common-core-facts. She mentions a specific change in the standards for which she was directly responsible. You make a distinction between the initial drafting and the feedback, but it didn’t work that way. The two went hand-in-hand, with many cycles of drafting and feedback and redrafting and further feedback.
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Cathy,
I look forward to your analysis of the Common Core.
There are a number of issues that come into play, in my view.
First of all, there is the unusual way that these standards were drafted — in secret, with little participation by K12 educators. They were then “adopted” in many states with little input from educators or the public. This was brought about by coordinated actions of the Gates Foundation and the Department of Education. Some defenders of the Common Core wave this off, as if all that matters is the content of the standards. But the content of the standards reflects its origins and intent. And I think it sets a dangerous precedent to allow a corporate philanthropy to essentially buy the process by which national education standards are set.
The second huge issue is the fact that these standards are deeply embedded in a standardized testing accountability paradigm. They were written to be tested, and the driving assumption is that all students must be “on grade level” starting in Kindergarten, to ensure that they will emerge from high school “career and college ready.” Standards have been backward mapped from that assumption, leading to some topics being pushed down to grade levels where they are inappropriate. Early childhood educators have been especially concerned about this.
We have seen the impact of the new, “more rigorous” Common Core tests in New York, where they were given last spring. Only about 30% of the students passed. The numbers were much worse for African American students, and English learners and Special Ed were worst of all. When these results are used for promoting and retaining students, and teacher evaluation, and deciding to close schools, the results will be devastating.
Here is what I wrote a few months ago: http://blogs.edweek.org/teachers/living-in-dialogue/2013/11/common_core_standards_ten_colo.html
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My knowledge of the standards and related controversies comes mostly from newspapers and the web. Most opinions concern just one or two of these aspects:
a description of curriculum topics;
means to assess a student’s understanding of those topics;
teaching materials and training informed by those topics;
policies allocating classroom resources to those topics;
policies requiring periodic assessment of students;
policies that use summary assessment scores to rank students, teachers, schools and systems;
policies that give such ranks inappropriate emphasis in personnel evaluations or resource budgeting;
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It’s important to put your analysis into a larger context. I recall the initial fierce opposition to the Massachusetts MCAS. Subsequently, I talked with a number of teachers about the reality that they experienced. The context really matters.
First, the quality of school administration is crucial. MCAS could be used as one of many tools or as a gospel. When treated as one of many, it turned out to be very useful for teachers and students. Teachers quickly found it to be a good motivator for some of their students.
Second, the administration needs to have a good grasp of how to accomplish success. It helped that experience soon showed that the best and easiest way to pass the MCAS was to teach the subject. Teaching the test was harder and less effective. Once administrations grasped that crucial fact, the test could become a motivator to really teach the subjects well.
But, in the hands of incompetent leadership, it did badly.
I don’t know where the Common Core fits in context. I have heard lots of concerns about the differences between MCAS and Common Core, both that the early grades are unreasonably hard and the advanced grades are too easy and miss important subjects.
Another important context is what is the mechanism for improvement of the Common Core. What is it? How does it work?
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As a teacher in the trenches with no connection to the folks who write the standards or make decisions, I have been surprised and pleased with the Common Core Standards for my 7th grade classes. I don’t know how it has played out in other districts but I am in Berkeley California and we have been able to actually do some real math and think deeply about such things as what exactly a fraction is and what model is most useful and durable. Instead of a to-do list or the latest cute gimmick, we get to respect students’ intelligence by bringing reliable and rigorous math to the classroom that will hold up under the weight of subsequent learning and discovery. I am seeing all of my students delve deeper into math understanding than every before. I am excited and challenged by what is expected of me as well.
Liz
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I did some delving into the content of the standards, so I might be able to give you a headstart: http://jeremykun.com/2013/11/04/deconstructing-the-common-core-mathematical-standard/
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Here are some difficult questions which sit underneath discussions about the common core.
1. Our public education has a tripartite funding model of local, state, and federal dollars (approximately 45%, 45%, 10%, from what I’ve heard, with geographic variation). Since budgets are tight, and for regulatory reasons, school districts are unable to abandon any of these partners.
Question: Given that all three funders want to guarantee their money’s not wasted, what sort of accounting process is reasonable?
2. Currently, the standards cycle operates as follows, as far as I know: state and federal standards are created by large committees. As these standards are created, publishing houses create curricula — not just textbooks, but day-by-day scripts for some schools. Often a school district will pay a company to create these scripts for their teachers and administrators will check that the teachers is on the right page on the right day. Teachers like to follow scripts that make sense, and it takes a while for them to a adapt to a new script. Standards change, and experienced teachers know that the cycle begins again.
Question: What can we do to enhance the teaching labor force, or to establish trust in the teaching labor force, so that teachers are capable and trusted to lead a class without scripting?
3. What is the connection between the on-paper standards and the on-the-ground teaching, and does this eventually make the standards less relevant? For example, suppose the standards recommend something developmentally inappropriate, e.g. someone inserted “In first grade, students should learn how to graph quadratic equations.” into the standards. In practice, a good script-writer might make an activity where the kids threw balls in the air and drew pictures of balls being thrown. Then there would be professional development institutes on quadratic reasoning, and various ball-throwing activities, ipad apps to simulate ball-throwing since throwing stuff in the classroom is dangerous, etc.. At the end, the meaning of “graph quadratic equations” from the standards is far far different from what happens in the classroom.
Question: What is the connection between the standards, as written by a mathematically sophisticated group (hopefully), and the classroom practice? How closely should the classroom practice resemble the desires of the standard-writers? Should we monitor obedience to standards or monitor developmental appropriateness of standards, and how?
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Fair Test, critical of high stakes testing, has a position on the Common Core here:
http://fairtest.org/common-core-assessments-factsheet.
Bob Somerby’s blog (search: “common core”) has some critical positions, though
most focus on what might be called the “moral panic” of failing schools that’s not
supported by careful assessment of comparative international assessments. If the
problem is poorly scripted then there’s a question about the “solution.”
Aside from the question if there’s a sustainable rationale for improving education,
the Common Core approach again looks to be another add-on to the school house
and its teacher without: a) accounting for all the add-ons for teachers to address,
and b) the degree to which communities and the nation as a whole reinforce the use of critical subjects such as arithmetic, not to mention mathematics. A half century ago
commentators such as Paul Goodman raised provocative notions as to where education takes place in comminities though today many of those settings have been wiped out.
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CC cannot be understood as a good/bad thing. The most important contribution you can make is to emphasize there has been a big conflation of the many different effects and aspects of the Common Core:
– development process (private organization macho and yes not iterated in serious trials with input from many groups)
– written content standards (so so, many complaints about early grades)
– the written emphasis on math practices (very positive and a driver of a lot of district/teacher attention to student learning, conceptual understanding, and ability to do anything with math)
– the impact on textbook curriculums (to be seen, but seems to be freeing districts to try more reforms texts like CPM, so that’s good)
– the many districts moving to an integrated high school curriculum (a bit frightening given lack of resources to prep teachers)
– the impact on teaching practice (some promise but again woeful resources for PD during a too-short rollout)
– its use to drive high stakes testing (to be determined, testing was already warping and poisoning education; the practices aren’t easily testable, so probably the rote outcomes will be emphasized, bad)
– its use in dubious teacher added value type assessments (to be determined, but these were happening anyway)
– its use by people trying to destroy public education (to be determined, but these efforts were happening with or without CC)
– its potential in helping math educators collaborate across states (high, but to be seen)
So far, the effects of the CC on my work with teachers has been very positive. That’s because people are focusing on the practices. The other shoe is going to drop and that’s when the new assessments hit along with whatever crazy uses people have for the resulting data.
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Yes, look at the content. Look especially at the content where it jumps – spots where you have to make a turn to more difficult material and how students are prepared for that. But to me, the critical piece to think about is the end goal: what is the END GOAL of this education method, and how do they articulate it? “Readiness” is not a goal, it’s a valueless noun made out of a very vague adjective. What kind of mind is being shaped? What kind of learner is going to succeed, and what kind of learner will be completely marginalized? If we all have to learn exactly the same stuff in exactly the same way, that removes flexibility to address differences in how and how fast students absorb the material. Who wins and who loses?
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I think reading a wide variety of teacher voices on the issue including those with concerns (careful with those who inaccurately or strategically conflate issues) and those who support it (including the statements from several Teachers of the Year). In addition, analyze the policy positions of all the major mathematical organizations (NCTM, NCSM, AMTE, ASSM, and CBMS) who support the standards and serve teachers. Consider other teacher-related organizations like ASCD and organizations like Edutopia. When the standards appeared and certain misconceptions begin to rapidly occur, I offered my voice of support here: http://hechingerreport.org/content/there-are-no-miracles-but-there-are-teachers-an-educators-view-on-the-common-core_8045/
Will be interested how you take the best of both!
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One thing to be aware of when reading the perspectives of professional organizations like ASCD and others is that these groups have become very much like businesses themselves, with a huge range of products and services that they sell. They have made a political decision to support Common Core, which may have been influenced by the opportunities for their organization that this provides. Not to mention the direct grants they have received from the Gates Foundation for their support in promoting and implementing the standards. ASCD alone has received more than $3 million from Gates for Common Core related work. I wrote about this here: http://blogs.edweek.org/teachers/living-in-dialogue/2013/12/is_ascd_embracing_market-drive.html
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Are aware of funding for the Mathematical Organizations listed? I would be interested to know those as well.
I am aware of the funding ASCD receives and the services they provide, but see their support for the core as both confidence in the standards in and of themselves but also as a support for teachers who are attempting to meet the standards as states adopted them.
As others research topics in education how can we grapple with funding of initiatives? Is there a framework around which we can define appropiate funding, especially as it relates to work in the public sphere? I would be interested in knowing how we or if we can relate private and public ethically?
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If evil Herod had his way, Baby Jesus would have been murdered in Bethlehem with the rest of the babies. Instead, Herod’s rotten plans ended up with Jesus safely in Egypt: perfectly fulfilling scripture. My point here is that I don’t care who gets a free watch out of the deal. We need to stop stalling and start delivering math instruction that will prepare our students for STEM jobs employers are unable to fill while American young people are struggling to find viable careers without math skills.
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“STEM jobs employers are unable to fill”
Excuse me? US colleges and universities produce twice as many STEM graduates per year than there are new STEM job openings. Employers are not unable to fill those openings, they are unwilling to pay enough to attract qualified STEM workers in the US. The claim that they cannot fill those openings is propaganda to increase the number of foreign workers who are willing to accept lower wages.
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Propaganda? http://www.monster.com/
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Yes, propaganda. I have a friend who is fully qualified for myriad jobs, fills out the applications, and then he’s not even invited for the interviews. Many postings are to meet hiring requirements for jobs for those who need visas and work for substantially less pay.
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Dean Baker makes this point: “Claims of a shortage of STEM workers have been disconcerting to those of us who believe in economics since shortages are supposed to result in rising prices, or in this case, higher wages. We don’t seem to be seeing rapidly rising wages in most areas, which makes the claims of shortages dubious.
“It turns out that at least one major tech firm has figured out how markets work. Netflix apparently doesn’t have any problem hiring STEM workers. It offers higher wages.” ( http://www.cepr.net/index.php/blogs/beat-the-press/netflix-overcomes-the-shortage-of-stem-workers )
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Jared Bernstein also chimes in. 🙂
http://jaredbernsteinblog.com/i-cant-find-skilled-enough-workers-at-the-crappy-wage-im-offering/
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I’ve been teaching high school for 10 years, NBT with 30 hours of grad math, PAEMST finalist. I have studied the K-8 standards and use them to remediate high school students with fantastic results. To improve my high-school-level instruction, I’ve watched Phil Daro videos and the Teaching Channel to learn how to use strategies to support the CCSS “explain” standards and my students are retaining much better. I make deals with them, “I won’t lecture if you will put for the mental energy to analyze this problem.” Class time is much more productive these days. Before the advent of the CCSS, I had talked with MANY teachers who didn’t think we should expect students to have much conceptual understanding. Reading comments in this blog, I see one extreme to the other. We have J2kun who believes we should be giving problems to 3rd graders that have no solutions. Then we have the other end that think CCSS tests will (because they are too hard) make kids cry. This makes me very confident the CCSS are a good balance of rigor. The standards give our kids a realistic chance they can beat the (current) single-digit odds of graduating with a high-demand STEM major in the usual 4-5 years of post-high work. For the standards to do their work, though we need to: nixthetricks.com
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I couldn’t agree more. Thank you for encouraging me to head into classes this week with explanation, conceptual understanding, and fluency in balance.
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This is must reading…among the many sobering reports emerging from sites like Edge.org’s annual invitational — this year’s query is “what scientific idea is ready for retirement?” http://www.edge.org/annual-questions — was this reposting on West Coast Stat Views from a Gelman post of a 60s anecdote by Richard Feynman recounting his experiences on a Los Angeles Board of Ed math textbook selection committee.
As noted in the lead-in to Feynman’s droll report, “Given the striking parallels between the Sputnik/New Math and PISA/Common Core relationships…”
The sad thing is that things haven’t changed all that much…
http://observationalepidemiology.blogspot.com/2014/02/judging-books-by-their-covers-by.html
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Here are some of the reasons I have found the Standards to be hugely promising:
1) Carefully built conceptual understanding of fractions
2) Addition is solidly related to linear measurement; multiplication to area
3) Multiplication is also associated with scaling
4) Multiplication naturally connects with proportional reasoning through which students can understand y=kx instead of “setting up and solving proportions,” thus avoiding the questions, “Do I multiply straight across or cross multiply?” “Do I need to ‘put a one’ under there?” This also connects better with math used in their science classes.
5) Careful attention to place value understanding
6) Mathematics is presented as a symbolic way to solve real-life problems. Because of a narrow focus at each grade level, students can be taught to understand the reasoning processes, rather than memorize a battery of formulas.
7) High school quadratics focus on modeling and completing the square (vertex form) instead of memorizing x=-b/(2a), opening the way for problem-based-learning strategies (PBLs).
In order to better understand the vertical alignment, I have outlined how area models in the Standards can be extended to connect from kindergarten through completing the square in Algebra: http://tinyurl.com/areamodels
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“2) Addition is solidly related to linear measurement; multiplication to area . . .
“4) Multiplication naturally connects with proportional reasoning through which students can understand y=kx instead of “setting up and solving proportions, . . .
“6) Mathematics is presented as a symbolic way to solve real-life problems. Because of a narrow focus at each grade level, students can be taught to understand the reasoning processes, rather than memorize a battery of formulas.”
Thank you. That sounds promising. I know that when I was taught arithmetic, it was just a matter of learning the rules, rules that did not have much to do with anything. No wonder so many people become math phobic. As Piaget might say, concrete operations come before formal operations. It sounds like the Common Core is giving elementary school children a good grounding in mathematical concepts.
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Cathy, the only way to evaluate the Common Core is to dig deep into a subject area, let’s say High School Math, and see if you agree with the Common Core or not. My evaluation or so-and-so’s evaluation is not enough to go by, as I can only tell you my biases. For example, do you agree with lanewalker2’s comment about not memorizing x = -b/2a…? You can wake me up at 3am and I can solve HS math problems rapidly and easily because some formulas are embedded in my brain. Is that good? Is that bad? Only you can say.
It’s worth noting that the Common Core for Math was not put together by math educators, but by educrats – people in Schools of Education – not specialists in Math, but in “Education” generating the ever revolving of standards which never produce the improvements promised.
My two bits.
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Good point. And I would explain the shift from memorizing x = -b/(2a) as important because it allows more time for curve fitting in general (focus). Shifting a vertex form of a parabola connects to shifts of families of functions in general. I have not seen “axis of symmetry” related to absolute value, yet certainly there is one. So we have ended up teaching quadratics as a mostly compartmentalized subject. When the focus is on modeling as in CCSS, the students are learning skills associated with STEM careers: digital signal processing, GPS programming…to name two fairly new ones. Actually linear, exponential, and logarithmic data appears more frequently in the sciences; but if taught as a study of families, quadratics are worthy of inclusion. Linear and exponential have priority in the standards, as they should; and as with the CCSS, quadratics can easily be connected if not taught as memorized formulas.
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I don’t get it when you say CC for math “was not put together by math educators, but by educrats – people in Schools of Education – not specialists in Math, but in “Education” generating the ever revolving of standards which never produce the improvements promised.” The three main writers of the math standards are Bill McCallum (a mathematician), Jason Zimba (who holds PhD in both Math and Physics, I believe), and Phil Daro (a former district math supervisor). They did consult mathematics education researchers and psychologists.
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Phil Daro is the perfect example of an educrat. BA from Cal in English. Teaching Certificate and one time teacher. Makes his living not from math nor from teaching math, but as a “consultant” on math standards. And when you look at all the committee members on Math, many are like Mr. Daro, even if there were some Math professors involved.
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I don’t care if Phil used to teach poodles to sing. I’ve learned so much by watching his videos that it’s really hard for me not to gush. Suffice it to say my classroom is a dramatically stronger learning environment.
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“For example, do you agree with lanewalker2′s comment about not memorizing x = -b/2a…?”
Well, I memorized that formula. But I also took satisfaction in knowing that I could derive it if I needed to, and did not have to rely upon memory. Knowing how to derive it was much more important to me. Still is. 🙂
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Yes, I can derive it, but at 3am I don’t need to derive it, just solve it. Both important skills. And for many students who cannot derive, at least they are left with a working method in their toolkit.
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These are really good things to talk about, and I’m really glad we have places like this to do so. The issue we are grappling with is how to equip students to be able to apply mathematics in problem solving. I think we all agree that for most math textbooks through College Algebra, the word problems are not the focus. I believe that in most math classes, it is possible to never solve a word problem correctly and still get a B. Maybe it’s better than that now….? Anyway, we want to improve problem solving skills. In order to do that, we will need time to focus on a smaller, stronger pile of foundational, extensible skills (long division is more versatile than synthetic, etc.). Memorized formulas are quick, but they do not help develop the analytical thought processes we are aiming for unless student is able to derive and modify them to fit various contexts. The “working method” of finding the axis of symmetry that would connect well with other concepts and inspire analysis is completing the square to get the vertex form of the quadratic.
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“long division”
They still teach long division? Gag me with a spoon!
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Would you agree that there are two type of students, those who will become mathematicians, and those who will (just) use math? If you do agree, then shouldn’t we differentiate between what an aspiring mathematician needs to learn and what an aspiring non-mathematician needs to learn? Being able to derive formulas is a very useful skill, but if a student, for whatever reason, cannot derive, don’t we at least want to leave her/him with some math skills for solving engineering, finance, biology, chemistry or economics problems?
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Another great question. Mathematics programs are frequently separated into “pure math and applied math.” The pure is mostly writing proofs and the applied more industry-related. However, I would disagree that Algebra I, for example, should have one class that memorizes formulas and plugs and chugs numbers into them and another class that learns to solve problems. There are some formulas like Pythagorean Theorem that will only be loosely understood by young students; but,in the last 10 years, I’ve realized the students can understand most math concepts very well. I work with many kids who are below grade level when I get them, and they are usually at or above grade level by the end of the year as they quit “putting” and start visualizing and making sense of the “steps.” You can imagine how much happier my students are exploring parametric changes in vertex form while fitting a quadratic to an interesting data set plotted on their graphing calculator instead of memorizing x=-b/(2a) and asking, “When are we ever going to use this stuff?” Both aspiring mathematicians and non-mathematicians benefit from sense-making. I am not suggesting engineers need to be able to prove all of Laplace, but for young children, understanding is far superior. I see it in 14-year-olds every day and often hear from parents and teachers of elementary students who are amazed at how their students are able to explain fraction operations having learned from Singapore math or other CCSS-type instructional materials. Every year, as far as we can remember, we have lost a huge percentage of Algebra 2 students in second semester because they can’t handle fractions. Under the CCSS, I am very optimistic that will change.
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When my girlfriend was in grade school the teacher had her add 48 to some number in front of the class. She got the right answer, and then the teacher asked her how she did it. She replied that she added 50 and then subtracted 2. The teacher then told her that she was doing in wrong. My girlfriend was a better mathematician than the teacher.
Learning a mechanical method may let you find the right answer, but calculators are faster and more accurate. Understanding is more important. My girlfriend understood that adding 50 and subtracting 2 is equivalent to adding 48, even though no one taught her that. The teacher also understood that, but thought that the method in the textbook was more important.
A friend of mine with an advanced degree in psychology decided to go back and study some math, which she had not done well at in school. She asked me to help her. I thought that we would be doing algebra, but she started with fractions. A lot of people never really get fractions, it seems. She was struggling with reducing fractions to the lowest terms. I remember that a lot of kids had trouble with that in school. I showed here the Euclidean algorithm for finding the greatest common denominator. She got the idea right away. Reducing fractions went from something that was difficult and uncertain to something that was easy and fun. She asked, “Why don’t they teach that in school?” You might well ask. Maybe they do now, I dunno.
When John Conway visits an elementary classroom and teaches the kids something, they always have fun. And why not? Isn’t that better than deadly drill?
Few students will become professional mathematicians, just as few will become professional musicians or artists or dancers. But they can become amateur mathematicians or musicians or artists or dancers. And they can gain the understanding to appreciate math, just as they can appreciate music, art, or dance. (There is a connection between mathematical ability and musical ability, by the way.)
Also, as lanewalker2 says, math develops problem solving skills, analytical skills, and abstract thinking. Those skills will stand students in good stead throughout their lives, even if they do not remember the formula for the volume of a sphere.
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The biggest problem with the common core standards is that they are linked to a particular “grade level” which, in turn, is linked to a particular age. I think that if schools abandoned age-based grade-level cohorts, which was introduced in the 1920s as part of the factory school paradigm, the common core would be welcomed as a means of logically sequencing the math curriculum. Because we now intend to use standardized “common core” tests of students in age-based cohorts as a basis for teacher evaluation the arguably valuable use of the common core as a sequencing tool has been lost.
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There is a great deal of truth to how age shapes our educational system, and I have seen how districts have been forced to be creative to address this and provide the learning experiences students need regardless of age. For some students it may work, but it cannot possibly serve every student (few things can). I do think the “progressions” within the k-8 standards can be lifted from the age boundaries and be used to navigate the development of ideas over time at whatever pace and serve as trajectory of learning in and of themselves.
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Yes, and here is the list of “must see” (CCSS chief writer) Phil Daro videos that explains what this looks like in the classroom. I provide links and brief summaries here: https://dl.dropboxusercontent.com/u/7405693/Videos/Phil%20Daro%20beyond%20coop%20groups.doc
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This is true IF we don’t use age-based tests to determine their progress… and I think that if technology has ANY role in education it is to facilitate the personalization and individualization of schooling.
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I’m thinking there are things I don’t understand about age-based tests that you might be able to explain here. We have grade level expectations. We don’t expect every student to meet that level and we try to draw as many students toward the advanced level as we can. To me the “age” or “grade” expectation is helpful indicator that a student might need extra support or challenge. One of my biggest joys is seeing how many of my “below-grade-level” students are at grade level (or advanced) by the end of the year. What am I not seeing?
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What you might be missing is that age-level tests are not always used as general guidelines, but are increasingly used as solid hurdles that students must clear to advance from one grade to the next. They may even be used to determine whether a student is promoted from one grade level to the next. This creates tremendous anxiety for students.
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Oh, wow. I’m glad I don’t work in a district like that. We use multiple ways to assess what our students need and have a whole “pyramid” of interventions. What districts are doing such things? Maybe they need to be called out…? Thanks for explaining.
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“Grade level” is a statistical artifact… all of your children are advancing at SOME rate and, given sufficient time, all of them should be expected to achieve the ultimate level of performance… our assumption that everyone will advance at the same rate is absurd given that student’s intellectual maturity is as variable as their intellectual maturity…
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Expecting everyone to advance at the same rate doesn’t even work with adults. Smarter Balanced uses adaptive tests for that reason. I don’t know of anyone who thinks everyone will advance at the same rate. Certainly CCSS writers don’t. Phil Daro’s video lessons begin with the assumption there will be a wide range of abilities within a classroom he explains how to challenge every student where they are and bring them farther. I’m thinking because there are grade-level goals (like being able to walk six blocks), some folks think that is what every student expected do and that is all they will do. Some will not quite get there, some will run, some will create complex cognitive maps as they go the distance and beyond.
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I worked with middle school math teachers for several summers on the geometry part of the core standards. A couple of observations just to add to what has been already said:
(i) most of the teachers were excited by and supported a more conceptual understanding of the material.
(ii) If we are going to have high-stakes standardized assessments (which I oppose) it’s not a bad idea to have a national test, on the idea that it might be better written. As many as one quarter (!) of the problems on the state assessments are so badly worded as to be incorrect. Typical example from nj: arrow pointing left and arrow pointing right are shown. Which of the following transformations changes the first arrow to the second: (a) rotation (b) dilation (c) reflection. All of the above was not a possible answer. My daughters have had many similar questions on their exams. Badly worded questions do not affect all students in the same way: some students and ethnic groups are better at “guessing” what the exam writers really want than others.
(iii) On the other hand the ccss are somewhat naive when it comes to the relationship with assessments and actual lesson plans. They are very vague on many points. For example, they say that students should understand transformations of the plane, but they don’t specify e.g. whether students should know how to find a rotation
by angles other than multiples of 90 deg, and in fact many
curricula are just doing rotations by 90 deg around zero in coordinates. If students can’t find the rotation of one point around a center by an angle, of say, 30 degrees clockwise, do they really understand rotations? To give another example, ccss says that students should know shapes, but they don’t provide an actual precise vocabulary, e.g. is a rectangle a trapezoid or not? I understand their point of view but it would really save a lot of time if they would just provide some definitions. To give a final example, they say students should know the formula for the volume of a sphere, but what’s the point if they don’t understand anything about why it’s true? (One doesn’t need calculus by the way.) It goes against the spirit of the standards, and I have no idea how PARCC is going to test it in a meaningful way.
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I understand why defenders of Common Core try to dismiss critics who conflate the standards with other issues like high-stakes testing and teacher evaluations. But what does it mean when the Governor of New York conflates them? In bashing the NY Board of Regents, Governor Cuomo makes his priorities pretty clear.
“Common Core is the right goal and direction as it is vital that we have a real set of standards for our students and a meaningful teacher evaluation system. However, Common Core’s implementation in New York has been flawed and mismanaged from the start.
http://www.nydailynews.com/blogs/dailypolitics/2014/02/gov-cuomo-rips-board-of-regents-common-core-changes
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Here is something I wrote in another forum a while back:
Reading Education Professor Boaler’s paper I had to ask myself if a student doesn’t know what congruent is, how likely is that student to know what Rotation, Reflection, Translation and Dilation are? … So I went to the Stanford website and found that Math professors think Boaler’s research isn’t worth the paper it’s printed on.
ftp://math.stanford.edu/pub/papers/milgram/combined-evaluations-version3.pdf
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Interesting. One of the these writers (Miligram) has been traveling around the country repeating known misinformation about the CCSS. Sounds like Stanford is not exactly a bastion of integrity, but I guess we shouldn’t be surprised by that. Who can one believe?
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I assume you mean Milgram of Stanford Math
http://www.reasoningmind.org/biographies/bio.php?bio=17
and his dispute with Boaler of Stanford School of Education. The two have very different opinions about what math standards should be. But to impugn Stanford (as a whole) and say that “we shouldn’t be surprised by that” is rather unfair.
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Agreed. I didn’t mean that. The Country as a whole seems to be moving in very selfish directions. Where are bastions of integrity? Lots of churches are messed up too. Thanks for helping me to clarify a hasty thought.
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Since I’ve gotten a head of steam up: 😉
In elementary school my girlfriend understood that you can add 48 by adding 50 and subtracting 2. I expect that that understanding was within the cognitive capabilities of most or all of her class. To echo my psychologist friend, “Why didn’t they teach that in school?” It is no secret that students stumble over carrying the one. They start to develop math anxiety. Instead of teaching, “4 + 8 = 12, write down the 2 and carry the 1,” why not teach, “4 + 8 = 4 – 2 + 10 = 2 + 10?” That makes sense. It furthers understanding. In fact, why not teach the soroban (abacus)? Subtracting 2 and adding 10 is normal on the soroban.
And adding on the soroban involves concrete operations. Remember, concrete operations before formal operations. I remember in grade school students were discouraged from adding on their fingers. Why shouldn’t they? Why the hell not? Concrete operations before formal operations. (BTW, the Japanese have a neat way of counting with their fingers. Start with an open hand. For 1, bring your thumb to your palm. For 2, bring your forefinger down, too, over your thumb. For 3, add your middle finger. Etc. 5 is a closed fist. Now what? for 6, raise your pinkie. And so on. 10 is an open hand again. Yes, it is ambiguous. Is that a 7 or a 3? 😉 You have to keep track.)
I mentioned showing the Euclidean algorithm to my friend. (I wrote “denominator” intead of “divisor.” My bad.) Actually, that’s not exactly what I did. What I did was to ask her if a number went into two other numbers without a remainder, would it do the same for their difference? Of course, she replied. How do we find such a number for 9 and 6? OK, now reduce 6/9 to the lowest terms. Etc. If I were writing a textbook, for reducing fractions to lowest terms I would start with problems like this. “Reduce to lowest terms: 6/3, 3/6, 9/6, 6/9,” with the fractions together in that order. Some students would discover the algorithm (or a variant) for themselves. Let them teach the others. Is the Euclidean algorithm too difficult for the students? Then I submit that fractions are also too difficult. When I was in middle school, most of my classmates struggled with reducing fractions to the lowest terms. It seems like the teaching methods of the time were guaranteed to produce math anxiety.
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Now for a couple of fun topics. At least, I found them to be fun when I was a kid.
Ever hear of Austrian subtraction? Salesclerks used to use it to make change. Suppose that an item costs $1.17 and the customer hands you a dollar and a quarter. What you do is say, “One seventeen,” and give them three pennies and say “One twenty.” Then you give them a nickel and say, “One twenty-five.” (Or “One and a quarter.”) You subtract by adding. Pretty neat, eh?
How about Russian multiplication? All you have to know is how to multiply and divide by 2. And add, of course. Here is how to multiply 6×4.
6×4 = 12×2 = 24×1 = 24
Just multiply one number by 2 and divide the other by 2, until you get to 1. Well, that works fine if you are multiplying by a power of 2. Let’s switch the numbers.
4×6 = 8×3
Now what?
8×3 = 16 x1 + 8 = 16 + 8 = 16 – 2 + 10 = 24
How about this?
5×7 = 10×3 + 5 = 20×1 + 10 + 5 = 35
Got it? I thought so.
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Isn’t that more fun than memorizing multiplication tables? 🙂
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Both are needed,. Memorizing multiplication tables can be a blast in a high-energy classroom: http://www.autofixinfo.com/EFKzOHoWDoE0FE/multiplication-song:-12-times-tables.html
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Many of our “top” math students are there because they memorize steps well. My most unhappy campers are the ones that tried their best but got the steps and rules majorly confused. Many of my unhappy campers are freaking-ready to buy motor homes when they start making sense of the concepts and connecting them. The hardest part is changing their mindset from remembering “what to do – where to put” into reasoning about equality. Few (if any) of my HS freshmen this year would be able to wrap their minds around Euclid. Instead they would try to memorize the steps. But I did get a lot of them to show that when they reduce (15x^6y)/(21xy) using prime factors: [(3)(5)xxxxxxy)][(3)(7)xy] where x/x = 1, etc. they are “cancelling” the GCF and the resulting numerator is relatively prime with the denominator. That is an important connection I haven’t seen in Algebra 1 textbooks. I am optimistic that with 5 years’ prior learning from well-implemented CCSS, Euclid would be within cognitive reach.
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Here’s hoping that things click for your unhappy campers. 🙂
Fortunately, they have a basis for understanding equality from gaining an understanding of the conservation of some physical quantities at around age 8 or so.
More power to you, lanewalker! 🙂
Yes, previous learning is very important.
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Clarification. My unhappiest campers generally have huge “buy-in” when they start seeing all the connections. What is surprising is how fast the turn-around can take place.
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Right. The light dawns. 🙂
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