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Recent Articles & Reports

Peg Tyre’s latest article, “The Math Revolution” in The Atlantic

Peg Tyre writes about education and usually makes very good points.  I feel, however, that she missed the point in her latest article in The Atlantic.  The story is premised on exploring why there has been a surge in the number of teenagers who have excelled in advanced math topics, as evidenced by increasing numbers who have won prizes in prestigious math competitions such as the Math Olympiad.

 It is a worthwhile question to ask, because there are also many students in the U.S. who are NOT doing well in math, who must take remedial math classes in college due to lack of facility in basic, foundational skills, and procedures.  So it is logical to ask what are the kids who excel at math doing that is different from those who we frequently read about in newspaper articles.

 

 A One-sided View of the Extracurricular Schools

 She focuses on the extracurricular schools that these students attend, one of which is the Russian School. Students start early in such schools—as early as first or second grade—and continue on.  (Other venues include the website “Art of Problem Solving”. Which she mentions).  In talking about the Russian School, she talks with the founder of the school and his experiences with his kids when they were in second grade in Newton MA, in 1997. 

 I’d look over their homework, and what I was seeing, it didn’t look like they were being taught math,” recalls Rifkin, who speaks emphatically, with a heavy Russian accent. “I’d say to my children, ‘Forget the rules! Just think!’ And they’d say, ‘That’s not how they teach it here. That’s not what the teacher wants us to do.’  That year, she and Irina Khavinson, a gifted math teacher she knew, founded the Russian School around her dining-room table.

 

 Although this makes for a compelling story, it leads the reader to make assumptions, plugging in to what readers have read before. One assumes the teacher was giving instruction on how to do a particular problem or calculation. We are then told by the founder of Russian School that “it didn’t look like they were being taught math.”  In fact, we really don’t know what his kids were being taught. We do know that it was 1997, and for those knowledgeable enough, the NCTM standards were published in 1989 and were having considerable influence in how math was being taught in early grades throughout the U.S.  Could it be that his kids were experiencing the wave of math reform that was sweeping the country in the wake of NCTM standards? Some of the alternative strategies to standard algorithms that we read so much about (and which are attributed to Common Core) were part of this movement.  But we don’t know that for sure—in fact we only know that they were being taught to do something step by step, and Rifkin admonishes them to forget the rules and just think. The average reader, having been told by countless newspaper articles that depict traditionally taught math as rote memorization with no understanding, may assume then that this is what was happening. They are also led to believe that conceptualizing mathematical problems is something you can do without benefit of instruction or foundational skills and memorization.

 

A Brief History of Procedure vs Understanding and an Implicit Agreement that Memorization Obscures Understanding

 She does acknowledge that fluency is important and says “Teachers at the Russian School help students achieve fluency in arithmetic, the fundamentals of algebra and geometry, and later, higher-order math. At every level, and with increasing intensity as they get older, students are required to think their way through logic problems that can be resolved only with creative use of the math they’ve learned.”

 But then she poses the dichotomy between understanding and procedure that also has been written about in many newspaper articles and give a brief description of the prevailing arguments about how best to teach math:

 Fiery battles have been waged for decades over what gets taught, in what order, why, and how. Broadly speaking, there have been two opposing camps. On one side are those who favor conceptual knowledge—understanding how math relates to the world—over rote memorization and what they call “drill and kill.” (Some well-respected math-instruction gurus say that memorizing anything in math is counterproductive and stifles the love of learning.) On the other side are those who say memorization of multiplication tables and the like is necessary for efficient computation. They say teaching students the rules and procedures that govern math forms the bedrock of good instruction and sophisticated mathematical thinking. They bristle at the phrase drill and kill and prefer to call it simply ‘practice’.”

 It would be informative if she had identified the “well-respected math-instruction gurus” ; there are some such as Jo Boaler who although well-respected in some circles, are held in extreme disdain by others. The article perhaps cannot sustain that kind of detail, but by leaving it at “well-respected” she seems to settle the question once and for all, and the reader implicitly accepts that these people must know what they are talking about, when in fact there have been serious and valid concerns raised about the theories of these “gurus”.  In fact, current thinking by some of these experts is that memorization of not only procedures but addition and multiplication facts, obscures student understanding. 

 It is not enough to memorize that 6 + 9 = 15, students must also be able to “make tens” and express it as 6 + 4 + (9-4) or 10 + 5; thus “understanding” has now become part and parcel to memorizing number facts. Making 10's is fine but shouldn't be obsessed over. It's a next-to-trivial skill that should be moved on from, it is in service of other things. In many if not most cases, it serves as a memory work-around that unfortunately is being described as "deep understanding". 

 She does mention there are those on the other side who maintain that memorization is essential for building upon.  In light of these disputes, and what is happening and has been happening in math education for 2 ½ decades, it would have been extremely valuable and instructive if she had asked the opinions of the heads of these various schools where they stood on the matter.  But it is left only as a brief paragraph—yes disputes exist--and leavea it to the reader, to decide which one is correct. In the context of the rest of the article, the reader is led to believe that in the supposed dichotomy between procedure and understanding, understanding always should take precedence—which has been met with disastrous results as more enlightened readers know.

 

Brief History of Common Core and Professed Ignorance About Any Effects it is Having 

 She then proceeds to talk about Common Core and what effect that might have on math education. She is now talking about the population at large, not just the subset attending the special schools.

 "The Common Core State Standards Initiative walks a narrow path through that minefield, calling for teachers to place equal importance on “mathematical understanding” and “procedural skills.” It’s too early to know what effect the initiative will have. To be sure, though, most students today aren’t learning much math: Only 40 percent of fourth-graders and 33 percent of eighth-graders are considered at least “proficient.” On an internationally administered test in 2012, just 9 percent of 15-year-olds in the United States were rated “high scorers” in math, compared with 16 percent in Canada, 17 percent in Germany, 21 percent in Switzerland, 31 percent in South Korea, and 40 percent in Singapore."

 It may indeed be too early to tell what the effects of Common Core are, though there have been many indications in the papers and other media that many parents are not happy with the type of instruction students are receiving—and performance on SBAC and PARCC tests was low.  She points out that US students have not faired well. She cites states for 4th and 8th graders in terms of proficiency but fails to say whether this is NAEP, SBAC, PARCC or other exam. She then talks about international testing.  All well and good, but this trend of poor performance has gone on for many decades and there should be at least some mention that instructional practices have been dubious and subject to much controversy and protest.  Lack of mastery of foundational facts and skills is a key reason why some students do not advance very far in the subject. And from what has been written about Common Core, shows that it is being interpreted and implemented along the same math reform lines that have caused this lack of mastery in the first place.  Perhaps she felt that the article structure couldn’t support such a detail, but in my opinion it is quite important, and might even be one of the reasons (albeit not the only one) why more students have been enrolled in the special schools she is talking about.

 

Implication that Extra-Curricular Schools Don't "Waste Time" with Procedures and Memorization

 The dichotomy between memorization/procedures and concepts then comes up again, now in the context of these after school programs:

 "The new outside-of-school math programs like the Russian School vary in their curricula and teaching methods, but they have key elements in common. Perhaps the most salient is the emphasis on teaching students to think about math conceptually and then use that conceptual knowledge as a tool to predict, explore, and explain the world around them. There is a dearth of rote learning and not much time spent applying a list of memorized formulas. Computational speed is not a virtue."

 The narrative is now that these schools are all about conceptual learning and that’s been the missing ingredient.  Her choice of words “rote learning” and “applying a list of memorized formulas” is reminiscent of the typical mischaracterizations of traditional math that plague journalistic writing about education. It is not that simple. It may well be true that math could be taught with more challenging problems, but it is also true that without a good basic foundation of skills and procedures, students are not going to be able to use the conceptual knowledge as she has described it above. 

 

 Foundational Knowledge is a "By-Product" of Conceptual Understanding: Where the Priorities Lie 

Just in case readers questioned that perhaps foundational knowledge is more than rote, and contains significant conceptual understanding, she continues to paint such instruction in an unfavorable way, in talking about “cram schools”—those being different than conceptually-based schools like the Russian School.

 " “Cram schools,” featuring a mechanistic, test-prep approach to learning math, have become common in some immigrant communities, and plenty of tutors of affluent children use this approach as well, but it is the opposite of what’s taught in this new type of accelerated-learning program. To keep pace with their classmates, students quickly learn their math facts and formulas, but that is more a by-product than the point."

 In fact, the Russian School ensures that students are fluent in math facts and procedures. Again, it would have been informative had she asked teachers at this school how students learn the basics, and perhaps asked the students themselves. The above paragraph gives the impression that math facts and formulas come about as a by-product of conceptual thinking, rather than serving as the foundation for such thinking—again a reprise of the “understanding is essential for memorization” narrative.

 

Practice Takes a Back Seat to the End Product: Arias from Tosca

 But she spends her time exploring the conceptual part of the pedagogy as if that exists independent of the foundational skills:

 "The pedagogical strategy at the heart of the classes is loosely referred to as “problem solving,” a pedestrian term that undersells just how different this approach to math can be. The problem-solving approach has long been a staple of math education in the countries of the former Soviet Union and at elite colleges such as MIT and Cal Tech. It works like this: Instructors present small clusters of students, usually grouped by ability, with a small number of open-ended, multifaceted situations that can be solved by using different approaches."

And also:

 "Sitting in a regular ninth-grade algebra class versus observing a middle-school problem-solving class is like watching kids get lectured on the basics of musical notation versus hearing them sing an aria from Tosca."

  Problem solving is much more complicated than she indicates above. One doesn’t learn to sing an aria from Tosca by doing just that. It is based on years and years of training of basic vocal skills. The majority of music lessons are about skills. Musicality, like math problem solving, is built up from mastery of the basics. 

 

Problem Solving: Just Go Straight to the Non-Routine, No Need for Scaffolded Problems 

 She does provide some good examples of problems that students are given—and I note that one of them appeared in my 9th grade algebra book (the rope around the world, lengthened by about 60 feet, asking how high above the equator can the rope now be suspended) . I mention that I have seen this rope problem, because problem solving and how it is taught, is not some new arcane art. It proceeds by teaching students basic types of problems—and those would include distance/rate, mixture, work, number, coins, etc—which have frequently been derided by reformers as not motivating students to solve them, nor giving any significant problem solving skills. 

But they are an essential starting point, from which students are given variants of the problems, well scaffolded so that they ramp up in difficulty and graduate to non-routine problems. She does not mention any of this, but instead falls prey to the popular narrative that understanding and critical/conceptual thinking is obtain via open-ended problems and requiring students to solve using more than one method. This has become a cliché in education journalism, and unfortunately a way of life in math classrooms across the U.S.  There is a belief that continued exposure to difficult problems for which students have had little or no prior knowledge builds up a problem-solving schema, and provides them the motivation to learn what is needed to solve such problems in a just-in-time basis.  It is highly unlikely that the Russian School and other schools operate in such manner. But the article gives the impression that this is what is done, and feeds into the lie that unfortunately many educators believe and implement.

The Real Message: Let's Help the Gifted 

 There are other aspects of math education that are left out. One is the ability of the students themselves, though she does refer to “gifted” students. In fact, she ends on the note that perhaps we should be identifying those students :

 The No Child Left Behind Act, which shaped education for nearly 15 years, further contributed to the neglect of these programs. Ignoring kids who may have had aptitude or interest in accelerated learning, it demanded that states turn their attention to getting struggling learners to perform adequately—a noble goal. But as a result, for years many educators in schools in poor neighborhoods, laser-focused on the low-achieving kids, dismissed suggestions that the minds of their brightest kids were lying fallow. Some denied that their schools had any gifted children at all.”

 This paragraph ignores that for many students who in fact could benefit from accelerated learning, such students also need foundational mastery. Her solution is to better identify gifted students and attend to their needs. In fact, the gifted students have the same needs as the strugglers.  A question she should investigate is how many of the struggling learners may, in fact, benefit from accelerated learning if given proper instruction. She admits it is a noble goal to turn attention to struggling learners, but does not mention that the programs in place to teach math have been influenced by reform-math agendas and do not adequately teach the basics, despite the increased focus on helping the poor performers.  Use of Everyday Math and Investigations has not helped the struggling students; it has hurt them, and it has likely hurt some gifted students as well. 

 

How is Gifted Defined? Who Makes the Selection/Identification?

 Furthermore, it is not obvious how one would select gifted kids in the earliest grades for math potential.  Math is not an all or nothing proposition. Students either show promise by having a head start at home or they show some sort of “interest” which may not be meaningful. Success breeds interest and love and success in the earliest stages of anything has more to do with the mechanics than some deep understanding or analysis. Finding and separating these kids is not the role of educators. They should not attempt to separate those with promise versus those who just work hard. If they offer a proper (STEM-level) curriculum in class, push a little bit, and then provide advanced opportunities after school (in areas like math, robotics, poetry, whatever), then kids will get what they need—thus fulfilling what she calls “a noble goal”. Kids in a STEM level curriculum can join the after-school programs and be successful later on. However, if the after-school program provides required lower level instruction, then it’s just a divergent form of tracking. AMC math and the International Olympaid are competitions, not curriculum. The teacher should be the mentor and the one driving (and pushing) the learning process with a group of equal-level students. This should handle all required top-level learning. After-school should only be for “extra.”

 

 The "Gifted" are the Tide Who Will Lift All Boats

 Her message is one of “let’s help the gifted” and it is hard for the general public, particularly those who are new to her arguments, to argue with her good intentions. But so far, the way such mentality has been playing out, is to limit classes that could benefit many students to only those who qualify as gifted. A case in point is the limitation of algebra in 8th grade to those students who are deemed to be “truly gifted”, a pattern that is emerging in California and other states.  She is unwittingly creating a narrative of having the gifted community raise the tide that floats all boats.  Unfortunately, some students will still be failing no matter how much we sing the praises of an increase in achievement levels.

 She concludes with:

 "Perhaps the moment is right for members of the advanced-math community, who have been so successful in developing young math minds, to step in and show more educators how it could be done."

 I would agree that perhaps the techniques used in these schools could be extremely effective –particularly the mastery of basic skills and facts which serve as the foundation on which to build conceptual knowledge and problem solving skills.  But it could also be taken to mean more of the same math-reform philosophy of education: more open-ended, just-in-time learning, understanding taking precedence over procedure and memorization.  Given the direction math education has been going for twenty plus years, it is not hard to guess how this article will be interpreted.

  

 

Common Does Not Equal Excellent”: New Report Sheds Light on Deficiencies of Common Core’s Math Standards

 

 

The American Principles Project Foundation has just published a new report, “Common Does Not Equal Excellent.” Focusing on the K-8 Common Core State Standards for Mathematics (CCSS-M), authors Erin Tuttle and J.R. Wilson provide evidence that the CCSS-M’s dictation of an instructional approach blurs the line between standards and curriculum. The standards consequently undermine the professional judgment of teachers, whose task it is to know the varied learning needs and styles of their students. Tuttle and Wilson consequently refute the claim that the Common Core is benign, or “just a set of standards.”

 

The K-8 CCSS-M differ substantially from the standards of high-performing countries and are ultimately developmentally inappropriate. Leveraging topic coverage comparisons, Tuttle and Wilson demonstrate that the CCSS-M fail to embody the coherence and focus evident in the standards of high-performing countries. This failure stems in large measure from a focus on abstract-levels of cognitive demand and demonstration of understanding. In turn, such focus results in tasks that often stunt students’ learning. By contrast, higher-performing countries emphasize concrete levels of cognitive demand memorization, and procedural fluency. 

 

Tuttle and Wilson assert that the “rigor” claimed by CCSS-M is not so much in the content as in the expectation that students display knowledge—a task for which they are frequently left without adequate tools. Though cognitively heavy, the CCSS-M remain procedures-poor. Focusing too early on the abstract drives an insistence on inefficient computation strategies. More effective, proven, and developmentally appropriate methods are delayed by up to two years. For instance, the CCSS-M does not introduce the standard algorithm for addition and subtraction until grade 4. Standard algorithms for multiplication and subtraction are thus withheld until grades 5 and 6 respectively. These delays result in inadequate preparation of students for algebra and beyond.

 

In addition to the CCSS-M per se, Tuttle and Wilson also explore the K-8 Publisher’s Criteria for the Common Core State Standards for Mathematics as well as Progressions for the Common Core State Standards in Mathematics, the latter written by the CCSS-M’s lead authors. Both documents go into greater detail concerning the strategies and instructional techniques embedded within the CCSS-M, laying out expectations for textbook, assessment, and instructional alignment. These additional materials clearly inform the use of CCSS-M in the classroom, essentially determining inefficient instructional delivery methods that undermine professional teacher judgment. 

 

Tuttle and Wilson conclude there is no empirical support for either the claim or the expectation that CCSS-M will improve student achievement.

 

“Common Does Not Equal Excellent” has been made available in PDF format on the American Principles Project website. 

 

 Heartlander Series on "Common Sense Approach to Common Core"

 The Common Core math standards have leant themselves to interpretations and implementations along the lines of reform math ideologies.  This series explains how they are being interpreted, but also offers alternative interpretations that are more sensible and in keeping with a sequence of math teaching that is ultimately more effective.  Barry Garelick authored this series at Heartlander in 2014.  It is worth noting that Jason Zimba, one of the lead writers of the math standards showed appreciation for this series and agrees with Garelick that the standard algorithms in the CC math standards can be taught earlier than the grade in which they appear.  In one of his articles, Zimba recommends teaching the standard algorithm for multi-digit addition and subtraction starting as early as first grade.

Here are the articles:

 A Common Sense Approach to the Common Core. Part I

A Common Sense Approach to the Common Core; Part II: Third-Grade Fractions.

A Common Sense Approach to the Common Core: Part III: Dividing Fractions.

A Common Sense Approach to the Common Core: Part IV: Making Connections One Concept at a Time,

Who's to Say Teachers Can’t Modify Common Core? No One

 

 "Confessions of a 21st Century Math Teacher"

An honest and critical look at math education from the inside, from the author of “Letters from John Dewey/Letters from Huck Finn” For anyone concerned with what Common Core is bringing about in the name of 21st century math education, STEM education, and "21st century skills, this book is a must-read. 


"I am not an outright proponent of the philosophy that ‘If you want something done right, you have to live in the past’, but when it comes to how to teach math there are worse philosophies to embrace,” Barry Garelick explains as he continues from where he left off in his last book (“Letters from John Dewey/Letters from Huck Finn”). He describes his experiences as a long-term substitute teacher at a high school and middle school. He teaches math as he best knows how while schools throughout California make the transition to the Common Core standards. It is the 50th anniversary of key historical events including the JFK assassination and the Beatles’ arrival in the U.S. It is also the 50th anniversary of his first algebra course, the technical and personal memories of which he uses to guide him through the 21st century educational belief system that is infused with Common Core and which surrounds him. 

Adds Garelick: “ ‘Teaching Math in the 21st Century’ will never be required reading in any school of education in the United States. While this might be a great reason to read the book, it is also a shame because there is a serious lack of an honest discussion and debate on math education issues that really needs to happen in education schools and within the education establishment in general.” 

"The book offers a brief glimpse into the eye of the storm that matters to kids, parents and teachers: the classroom as it functions under changing curricula and mindsets and how stakeholders deal with it. The book shows how great teachers are desperate to deliver a solid education in spite of proclamations from disconnected, poorly-grounded leaders; it shows how students just want to learn math and parents want to feel confident and informed about the education their kids are receiving." Matthew Tabor, editor, Education News 

"If you want to know why a teacher without a political ax to grind and who is deeply committed to actual teaching would object to the Common Core--read this book!" David Olson, Asst Professor, Communications Studies, Southwestern University 

"It's difficult for those not involved in the daily scene of mathematics education at a K-12 level to understand exactly what it's really like on the front lines. Barry Garelick's book gives a day-by-day description, so you can truly understand the difficulties a math teacher experiences when teaching math the way it's supposed to be taught in today's system."  Barbara Oakley, author of "A Mind for Numbers"



(May 2011)

Why One National Curriculum is Bad for America - A Critical Response to the Shanker Institute Manifesto and the U.S. Department of Education’s Initiative to Develop a National Curriculum and National Assessments Based on National Standards  -

(over 100 signatories)

 

 

 Archived Articles & Reports

 
 
 
Fair to Middling? A New Pioneer Institute Report by Dr. R. James Milgram and Dr. Sandra Stotsky: http://www.pioneerinstitute.org/
 
New Report from The Pioneer Institute "Why Race to the Middle? Massachusetts and California K-12 State Standards Far Exceed National Standards Drafts" by Sandra Stotsky and Ze'ev Wurman.  
 
 
What are the benefits of a fast-track approach through high school? What are the possible problems and risks?
Numbers in mathematics education Wars: School Battles Heat Up Again in the Traditional versus Reform-Math Debate: Weak student scores fuel the fight
 
2.16.10 - Barry Garelick - “What’s a court doing making a decision on math textbooks and curriculum?” This question and its associated harrumphs on various education blogs and online newspapers came in reaction to the February 4, 2010 ruling from the Superior court of King County
  
Great post about the Barry Garelick commentary (see above)
Source: www.city-journal.org

 

2009 NAEP Math Scores 

  •   Full report found here: http://nationsreportcard.gov  

     

    Read the commentary from our co-founder, Barry Garelick, along with others in: 

      How to Improve National Math Scores - Room for Debate Blog - NYTimes.com

    read the comments too!

     Should All States Meet the Same Education Standards? 

        

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     How to Raise the Standard in America's Schools

    "Two-thirds of U.S. children attend schools in states with mediocre standards or worse."
     
     

     The Future of High School Reform

    Can we get to where we want to be on high schools without national standards? And if not, what do we need to do to get to some kind of common curricular standards in this country?

    MICHAEL COHEN: Well, let's talk about what kind of standards we need, whether they're national or not. It seems to me what we need in education are standards that are anchored in the real-world demand that students are going to face, that they reflect what you need to know in order to succeed in postsecondary education and in the workplace. They need to be internationally benchmarked as well, because our students are going to enter a global economy. They are going to be competing with young people all over the world. They need to be focused on what's most essential rather than filled with things that would be nice for students to learn somebody. They need to be vertically aligned so there's a logical, clear progression from what you start learning when you enter kindergarten to where you're going to end up at the end of high school, and they need to be assessed well.
     
     

     

     Click Here for More News  Around  the Nation

     Click  National Standards for Archived News and Information Regarding the Initiative for  

     
     
    Read our Mission Statement and information about the multi-state initiative currently underway, coordinated by the National Governors Association (NGA) Center for Best Practices and the Council of Chief State School Officers (CCSSO) called the  “State Common Core Standards”, a state-led process to develop voluntary "common core" K-12 English and mathematics standards. 

    Our Coalition is not "endorsing" national curriculum standards.  We are advocating that all appropriate stakeholder groups and academic experts be part of the process. We advocate for the involvement of research mathematicians and university math professors, in addition to the usual K-12 educators and professors of math education, in writing substantive math content standards that will prepare our children for success in college and life.

     

     

    Read our Coalition's comments on the CCSSI College and Career Readiness Standards here: U.S.CoalitionComments.pdf

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