Thursday, March 7, 2013

Common Core Elementary Math Standards

Let’s get this out of the way right now – the criticisms complaining that the Common Core Standards are the Federal government’s taking over of the schools - and the loss of local control - are completely moot.  That happened ages ago!  The 1990’s OBE/Blueprint 2000 goals for Reading, Math, History and Science were written by various committees at the Federal level.  That’s why the states all have such similar stupidities written into their current Math standards.  The states’ DOE’s took the Federal money along with the Federally written goals, tinkered with them enough to make them even worse and called them their own.  There was a huge uproar over Whole Language and the History standards so there have been rewrites of them, and Science is apparently still a battle ground, but the Math standards of today pretty much date to the horrific ‘90’s.  There are plenty of books and articles written about the history of education in this country which explain how and when this happened far better than I could.  What I’m doing here is examining these new standards from my perspective as a tutor, mother and grandmother in order to judge them on their own merits.

They Make It Sound So Good

At first I found myself liking the new Math standards quite a bit.  After the confusing, overwrought hodge-podge of current standards, they are almost poetic in their simplicity.  In fact, the authors mention their intent to correct the “mile wide/inch deep” errors of today’s goals.  If you click below to the 2nd Grade standards, for instance, you will see a much more realistic, pared down and far more grade-appropriate list of goals that are separated into four main areas of study.  

These standards appear much more successful at assigning reasonable goals arranged according to the logical, building block sequences that would be recognized as normal by most parents.  In fact, there is a chart in the “Publisher’s Criteria” section of Resources (pg 9) that lists topics that should not be assessed before certain grade levels.  Parents will be surprised to learn that Symmetry in Geometric shapes should not be assessed (nor taught, it should be assumed) before Grade 4, and Probability should not be assessed before Grade 7.  These topics are currently to be found at much earlier grade levels in many text books, and have served to be major distractions from what should be the main jobs at the Elementary levels.  The Core authors go to great lengths emphasizing what the main topics should be in the younger grades.

The misuses of number models such as those shown in my previous blog are also addressed:

“Research indicates that students’ experiences using physical models to represent hundreds, tens, and ones can be effective if the materials help them think about how to combine quantities and, eventually, how these processes connect with written procedures.” (Adding It Up, p. 198,…). For example, base-ten blocks are a reasonable model for adding within 1000, but not a reasonable method for doing so; nor are colored chips a reasonable method for adding integers. …. The word “fluently” in particular as used in the Standards refers to fluency with a written or mental method, not a method using manipulatives or concrete representations.”

It’s always interesting to come across evidence that the Experts have been running experiments on children instead of educating them.  The above is but one example of the Core writers citing research that that has been subjecting multitudes of students to an obvious, ineffective use of manipulatives and “models”, as well as several other instructional errors, that I’ve heard parents complaining of for years.  I’m sure that besides the report Adding It Up (published in 2001 by the National Research Council), there have been many other reports, books and articles published on this subject.  I wonder how many kids were labeled as learning disabled as a result of never being adequately taught the correct way to add and subtract just so the alleged “Experts” could find out what everyone else already knows?  But I digress.

The Core authors also criticize what became known as “aspiral Math” which covered the same material year after year.  The review, or teaching, of earlier material while also teaching grade-level skills is something I deal with on a daily basis as a tutor.  I’ve had so many students come to me needing help with fractions who do not know their multiplication tables nor, of course, their division facts.  (In the quote below, I see a problem with the expectation of proficiency with the standard algorithm for division coming as late as 6th Grade.  Plus, the problem with division is not a problem with place value knowledge in my experience.  I deal with this further in my “So Bad” section.)

“The basic model for grade-to-grade progression involves students making tangible progress during each given grade, as opposed to substantially reviewing then marginally extending from previous grades. Grade-level work begins during the first two to four weeks of instruction, rather than being deferred until later as previous years’ content is reviewed. Remediation may be necessary, … but review is clearly identified as such to the teacher, and teachers and students can see what their specific responsibility is for the current year.”

“… materials often manage unfinished learning from earlier grades inside grade-level work, rather than setting aside grade-level work to reteach earlier content. Unfinished learning from earlier grades is normal and prevalent; it should not be ignored nor used as an excuse for cancelling grade level work and retreating to below-grade work. (For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning about place value; this is more productive than setting aside division and backing up.) Likewise, students who are “ready for more” can be provided with problems that take grade-level work in deeper directions, not just exposed to later grades’ topics.” (pg. 12)

It is great to see the end of measuring using “non-standard” units of measurement which had some lesson plans wasting students’ time measuring large lengths – such as room perimeters - with paper clips or pencils – this explains so many kids’ lack of skill with rulers.  In the measurement and data sections, there is mention of learning the smaller units within the larger units and using this knowledge to solve various problems (i.e. there are 12 inches in a foot, etc).  It would be refreshing to begin meeting students who know measurement facts to the point where students could add and subtract units of time, length or weight, etc.  How long has it been since kids were taught how to add 2 yards, 1 foot, 6 inches to 3 yards 2 feet 10 inches and convert the sum to the proper units?  The authors also specifically mention telling time using both digital and analog clocks – many of today’s teachers are under the impression that children do not “need” to learn to tell time with analog (face) clocks.  

The solving of word problems is given a great deal of attention.  So many of today’s children have a deep fear of these problems, which inundate the FCAT and many other state tests, because they are not explicitly taught the “clue words” to look for.

“The language in which problems are posed is carefully considered. Note that mathematical problems posed using only ordinary language are a special genre of text that has conventions and structures needing to be learned. The language used to pose mathematical problems should evolve with the grade level and across mathematics content.”   (pg 17)

The authors address the need for a math curriculum to be fully balanced between the practical and conceptual aspects of math and spend a great deal of time and effort in explaining exactly what they mean.  They make distinctions between practice exercises and solving problems while stressing the need for both.  I included the second paragraph below primarily for their examples of “conceptual problems”.  Today’s standards also make much of bringing about a student’s conceptual understanding of math, but the actuality of what is presented under that category has been so detrimental to achievement, that the phrase is starting to make people cringe.

“To date, curricula have not always been balanced in their approach to these three aspects of rigor. Some curricula stress fluency in computation, without acknowledging the role of conceptual understanding in attaining fluency. Some stress conceptual understanding, without acknowledging that fluency requires separate classroom work of a different nature. Some stress pure mathematics, without acknowledging first of all that applications can be highly motivating for students, and moreover, that a mathematical education should make students fit for more than just their next mathematics course. At another extreme, some curricula focus on applications, without acknowledging that math doesn’t teach itself.

“Materials amply feature high-quality conceptual problems and questions that can serve as fertile conversation starters in a classroom if students are unable to answer them. This includes brief conceptual problems with low computational difficulty (e.g., ‘Find a number greater than 1/5 and less than 1/4’); brief conceptual questions (e.g., ‘If the divisor does not change and the dividend increases, what happens to the quotient?’);”…(pg 10)

This Is Going To Be So Bad

Once I noticed that the NCTM was involved in the writing of the Common Core Math Standards, I knew it wouldn’t do to assume the best intentions based just on the simple listing of grade level goals.  The NCTM (National Council of Teachers of Mathematics) wrote the current standards that have been wreaking havoc with American students for at least the past twenty years.  So, a-wandering I went through all the various sections:  the Key Points, the Standards for Practice, the Standards by Domain, the Resources – including the Publishers’ Criteria (very important) and so on. 

As it proved with the current standards, the authors do not often lie outright, but use misdirection, the distortion of definitions as normally understood, and very carefully worded statements designed to placate the reader (including me for a short, happy time).  The Experts are very talented and well-practiced at telling the public what they know we desperately want to hear.  It is such an old tactic - much that the Core authors have written will have us nodding in agreement so often that our inclination is to keep nodding and nodding in approval even after we sense that we should begin to question or object. 

The problems begin with their seeming to state that they have adopted the goals of countries well known for their prowess in math.  In fact," International Benchmarking" and "International Best Practices" are the by-words of this latest set of standards.  This would naturally lead us to believe that our students will be taught just like the Asian whiz kids are, but that is not exactly what the authors are saying.

“The composite standards [of Hong Kong, Korea and Singapore] have a number of features that can inform an international benchmarking process for the development of K–6 mathematics standards in the U.S. First, the composite standards concentrate the early learning of mathematics on the number, measurement, and geometry strands with less emphasis on data analysis and little exposure to algebra. The Hong Kong standards for grades 1–3 devote approximately half the targeted time to numbers and almost all the time remaining to geometry and measurement.” — Ginsburg, Leinwand and Decker, 2009

“International benchmarking provides an additional tool for making every state’s existing education policy and improvement process more effective, offering insights and ideas that cannot be garnered by examining educational practices only within U.S. borders.  State leaders can use benchmarking to augment their “database of policy options” by adding strategies suggested by international best practice to the range of ideas already under consideration. Indeed, international benchmarking should not be a stand-alone project, but rather should function as a critical and well-integrated component of the regular policy planning process.” (pg.23)

We are NOT getting the Singapore Math standards! The Core authors are almost clear in stating here and in other sections of the Common Core site that they have chosen only a select number of educational features of higher-performing countries which “can (not “will”) inform” the states’ process of developing K-6 Math standards.  They have opted to focus on the grade-level topics that these other countries target, but that does not mean that our public schools will be using any of the other academic features of these countries.  Culling inappropriate topics will do our children no good without all the rest that these other nations employ to assure their students’ success.  However, the Core folks clearly state in more than one area that the Core Standards do not adhere to International Standards to the point of mandating any particular curricula, materials nor, even more importantly, teaching methods which are very much a part of intensive teacher training and evaluation in the more math-proficient countries. 

“These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.”

The phrasing makes it seem as if the benchmarking in the USA is to act more as a reference than as a mandate to the states – somewhat interesting reading, in other words.  (And, really, The Feds aren’t allowed to mandate, are they?)  The most we can expect from the statements above is that the various states will be spending gobs of taxpayer money doing (allegedly their own) Standards rewrites that will include the phrase “International Benchmarking” a lot, but won’t be actually making much use of those much better International Standards.  This becomes obvious as we continue our reading.

Another very strong clue to the bitter disappointment we and our children will be experiencing with these Common Core Standards is the severe restriction in access to the Standard Algorithms for basic arithmetic operations.  Looking at the 2nd grade goal dealing with adding and subtracting, we find:

CCSS.Math.Content.2.NBT.B.5  Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”

This is misleading; the public will assume that this indicates the learning of the Standard Algorithm because that method uses place value as well as the relationship between adding and subtracting to solve problems. The truth lies in the distinctions the authors themselves apply in their use of language.  Accordingly, our public school 1st, 2nd and 3rd Graders are to be subjected to the same confusing, ineffective, multiple-strategies methodology that our students currently struggle with.  The actual wording: “Fluently add and subtract multi-digit whole numbers using the standard algorithm,” is not seen until Grade 4.  Multiplication using the Standard is delayed until 5th Grade, division until 6th Grade.  All four basic operations using decimals wait for the Standard Algorithm until 6th Grade.  The grade level sections on fractions – and all the “conceptualizing” that goes with them – I found very confusing as I searched in vain for any mention of the use of the Standard Algorithm.  Far from being balanced in juxtaposing procedure and concept, as claimed by the authors multiple times, the Core tilts so far toward “concept” for so many years that procedure is in grave danger of ever being achieved.  Surely many 4th Graders will have given up hope of ever performing the super complex function of - adding.  Are we really supposed to believe that the Experts think it takes four years to grasp the ideas behind adding and subtracting?

Much like today, students are also to try to figure out how to solve problems on their own without benefit of explicit instruction.  According to the Experts, this will take a great deal of perseverance on the part of the students – students must not be allowed to give up, which of course they would be pretty likely to do since they are not to be taught the skills nor knowledge it would take to solve them.

Under CCSS.Math.Practice.MP7  “Look for and make use of structure:
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, …”  

Very young students struggling to add - because of teachers insisting that they use ten different, inane methods to do so- might Not notice the above fact.  It would be far easier for a student to “notice” patterns such as this if taught the Standard Algorithm from the beginning, but that is not to be.  It would be even better if an instructor would point this out to the kids explicitly using several different examples which would naturally lead the students to find similar patterns on their own.  We wish.

 Below is another indication of the same as the authors distinguish between problem-solving and practice exercises.  The only thing students learn from trying to solve problems they lack the skills and knowledge to solve is frustration and a hatred of Math

“…. in working exercises, students apply what they have already learned to build mastery. Problems are problems because students haven’t yet learned how to solve them (Italics are mine); students are learning from solving them. Materials use problems to teach mathematics. Lessons have a few well designed problems that progressively build and extend understanding. Practice exercises that build fluency are easy to recognize for their purpose. Other exercises require longer chains of reasoning.”  (pg 17)

The most chilling statements I found on the entire web site, however, were these:

“…. each standard in this document might have been phrased in the form, “Students who already know A should next come to learn B.” But at present this approach is unrealistic—not least because existing education research cannot specify all such learning pathways (Italics are mine). Of necessity therefore, grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators, researchers and mathematicians. One promise of common state standards is that over time they will allow research on learning progressions to inform and improve the design of standards to a much greater extent than is possible today. (Italics are mine.)  Learning opportunities will continue to vary across schools and school systems, and educators should make every effort to meet the needs of individual students based on their current understanding.”

“These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that these standards are not just promises to our children, but promises we intend to keep.”

What kind of "lessons learned" would have these folks withholding skills and knowledge for even longer than happens now?  Put another way, why does this look like dumbing our kids down even more?  Because this isn’t about education – this is about the states getting big bucks to do yet more experimenting.

There are plenty of mathematicians, other education experts and no small number of parents who know exactly how to successfully teach math and in what sequence.  They know that the Standard Algorithms for the four major arithmetic operations can be successfully taught at grade-appropriate levels beginning with 1st Grade with no sacrifice to the understanding of the concepts of numbers or functions.  But apparently they were not the ones in charge of writing the Common Core Standards.  Despite all the yammering by everyone from President Obama on down about the importance of the STEM skills and knowledge, these Core authors are not in the least interested in educating our public school children.  A large percentage of our children are not to have access to a future in STEM-related jobs, and the United States is not to have access to the majority of its children's highest potentials.  The  Educators are far too busy doing research – and not in a way that will help this nation at all.  

There is, of course, even more being written on the Core’s errors in teaching Math at the higher levels which I’m not qualified to address.  There's quite a bit about the Language Arts Standards also.  I’ve read very good articles on this topic at Education  While some states have spurned the Common Core, many states have begun using the new standards, and the detrimental effects are being described in detail by experts and by parents of suffering children.  Please especially see: