As You Haven’t Been Told, Dept.

I teach math at a small Catholic school in California.  I teach 7th grade math and 8th grade algebra.  For those who have read my latest book, you know that I use a 1962 version of Dolciani’s “Modern Algebra” as my textbook.  The students like the simplicity of its presentation, and so do the parents. I have had parents tell me they like the book, and one in particular said that it is how she learned algebra, and it allows her to help her daughter. She thanked me, and said “I can’t stand that Common Core stuff.”

The Common Core stuff that irks parents are the alternative “strategies” that replace the standard algorithms. One of these is for multidigit addition/subtraction; another is for multiplication and division. As I’ve documented before, these strategies (such as ‘making tens’) are nothing new. Traditional textbooks of the past have taught them, but they were introduced after students mastered the standard algorithms.  The standard algorithms served as the “main dish” in the dinner party known as math. The alternatives were “side dishes” and the two were distinguishable. Now they are not; it is one big mess, with students sometimes thinking that they have to use a particular strategy for particular problems.

Therefore it is of interest to hear William McCallum’s view of this aspect of Common Core. He was one of the two lead writers of the Common Core math standards.  When I wrote an article that was published in the online Atlantic about Common Core, I pointed out that the standard algorithm for multi-digit addition and subtraction did not appear until 4th grade. Until then, teachers and students were saddled with “strategies” which included pictures and inefficient methods in the name of “understanding”. The view of reformers is that teaching standard algorithms first eclipses the conceptual underpinning of why the algorithms work as they do—this in spite of the pictorial explanations that appeared in early textbooks from the 60’s, 50’s and earlier that provided such explanation.

McCallum commented on my Atlantic article and disagreed with me that the standard algorithms were delayed. I provided him evidence until he finally stated that the Common Core standards do not prohibit the teaching of the standard algorithms prior to the grade in which they appear.  Specifically his comment was:

The standards (1) do not say that conceptual understanding must come first, and (2) also say explicitly on page 5 that ‘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.”

This was news to me, and apparently news that was buried in the material accompanying the standards, despite McCallum’s belief that it was made clear.  In particular, a guidance document for publishers, which came out in tandem with the Common Core standards, advises publishers not to test students on standard algorithms prior to the grade in which they appear in the standards.  I guess it’s OK to teach the standard algorithms earlier, but just not to ensure that students know them.

There are few “Common Core aligned” textbooks that address the standard algorithms prior to the grades in which they appear in the Common Core standards, so apparently McCallum’s word has not really made the rounds.  There is one exception and that’s the Common Core editions of Singapore Math. They do teach the standard algorithms earlier. They also test the students on them, which has cost them a penalty by the company EdReports which rates textbooks in terms of the degree to which they are aligned with Common Core.  Singapore Math’s Common Core edition is considered by EdReports to not be aligned.

Since people mistakenly believe that alignment with Common Core implies effectiveness, EdReports’ rating of Singapore’s books may have cost the company some sales.

The Standards continue to be interpreted in accordance with math reform ideology. And although McCallum in his remarks to me in the comment section of the Atlantic article stated that “the phrases ‘critical thinking’ and ‘collaborative learning’ do not occur anywhere in the standards and that the standards “neither dictate nor forbid any particular style of pedagogy”, the die has been cast for the lower grades (K-6).

In the world of Common Core, alignment equals effectiveness, and book publishers happily comply with the guidelines they have been given. In my opinion, as well as others, it would have been helpful if McCallum’s statements to me could have been made more public than in the comment section of an Atlantic article. It also would be helpful if publishers are not punished by outfits such as EdReports for ensuring that students know standard algorithms prior to the grade in which they appear in Common Core.

22 thoughts on “As You Haven’t Been Told, Dept.

  1. Ultimately, the fuzzies failures have to do with no enforcement of individual mastery at any level. Whether its traditional algorithms first or last, they fail to understand the deeper levels of understanding learned by doing and mastering lots of individual homework variations.

    In the early 60s, we never jumped right into any fixed algorithm. Everything was properly scaffolded. It’s ignorance or a fraud to say that it was any other way unless they cherry pick a bad implementation. However, the meme nowadays is that if students see different methods (not enforced by practice and mastery testing to any great extent), then the general concepts and understanding will translate to the details and mastery of the variations.

    Nope. It’s never been that way and it never will be.

    There is a lot of understanding buried deep in the mastery of the standard algorithms – and in all of the individual homework P-set variations that form the basis of traditional math – and that defines all proper high school and college math – and careers.

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  2. I’ve always reminded teachers that the CCSS are a floor. They are the LEAST you need to do to educate students. Not the best that you can do.

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    • Yes!

      It’s not STEM level.

      K6 schools need to be explicit about what mastery (slope) is needed to meet their 7th grade split to Pre-Algebra and then full Algebra in 8th grade. They need to AT LEAST publish their requirements/test they use. I saw kids (and parents) dumbstruck that they were not selected for the advanced classes after years of “exceeding expectations.”

      There is no magic understanding or engagement path that recovers from that. STEM careers are over by 7th grade without getting a lot of individual help from parents or tutors. All of my son’s STEM-ready friends in his high school AP Math sequence had help at home or with tutors.

      Another big dumbstruck event happened in 8th grade when “exceeding expectations” kids were warned to not to take too many honors classes in high school. What? Why aren’t they ready? Honors classes in high schools are like the old “College Prep” classes we had in high school. Now, College Prep is for everyone except those in the lowest levels because they expect all to go to college.

      There is a huge expectation slope change that happens in high school because K-8 has lowered their slope with CCSS (good for full inclusion dreamland in our schools). The onus is now on the students and parents to figure that out and deal with it. K-8 schools like to think that group engagement and concepts are all that’s needed when what kids need are lots of individual homework (facts and skills) and more feedback than CCSS rubric generalities.

      As one K-8 teacher told me, the best students manage or get help from parents, but the middle kids with parents not paying attention are hurt the worst. They are unprepared for high school and the real world.

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  3. My concern is that the Common Core Math Standards ask students to solve problems, especially in the early grades, in ways that cognitive scientists say emphatically that their brains are not able to manage. The reason has to do with the human brain’s “working memory limits.” I attempt to explain that, with neuroscience included, here: https://www.ChemReview.Net/CCMS.pdf . If the neuroscience gets to be too much at once that is unfamiliar, skip to what the experts have to say in English. The standards attempt to teach conceptual understanding before (and without) math fact automaticity. But at least look at the graph on page 2. — rick nelson

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    • “…the evidence indicates that a significant percentage of state math standards, now and for the past
      two decades, have asked students to solve problems in ways that a student (non‐expert) brain
      measurably cannot do. ”

      This is not some disagreement over what constitutes proper cognitive learning. This is all about low expectations hiding behind the cover of fancy learning talk.

      These CCSS fuzzies just don’t get the job done for mastery. Even if they agreed to teach the standard algorithms with no confusing alternative methods, they STILL would not get the job done.

      Proper learning requires a mastery/testing feedback loop that they just blow off. All proper STEM math classes in high school and college degree programs require the ability to do weekly individual homework problem sets. Mastery of those problem variations contain so much of the needed understanding – that these fuzzies just blow off. By the time their students get to high school, they lack mastery of the basic skills AND the ability to finish a homework P-set.

      I tutored many high schoolers who thought homework was something that you just made a stab at. Even after we had tutoring sessions where I got them to explain things in words, I told them to go back and do the homework themselves. Few did so. Understanding is meaningless unless you can actually DO the problems – and that graph shows the proof. However those results are based on far more than just the approach they are taking. It’s low expectations.

      When I was growing up, kids not only had the standard algorithms, we had homework and tests and report cards that gave understandable feedback to parents. If you didn’t get a fixed passing grade, you were subject to summer school or being held back a year. for my son in the 2000s+, his school gave out wordy three page rubrics that provided vague and meaningless feedback. You see the “exceeds expectation” rating and assume that all is great. All kids are promoted and all problems are pushed into the future as if learning is some natural process that requires no pushing at any level. These fuzzies do a great job making kids feel responsible for the results. One bright student I tutored in 8th grade almost screamed that she was “soo stupid!” No. It was years of low expectations, lack of skills, and lack of facts.

      Many of us parents, however, could see the lack of mastery and enforced that learning with things like flash cards (horrors!) and worksheets. I remember one of my son’s teachers being horrified when I
      happened to mention that I left out math worksheets for him that he loved to do. It took me a few years, but I figured out that this was NOT just a scientific difference of opinion about a cognitive approach, but low expectations and the dreamland hope that concepts and “thinking” and group work excitement will get you mastery of “rote skills” and “superficial facts.” Skills hide a LOT of understanding and facts are NEVER superficial.

      They expect kids to develop “grit,” but do absolutely nothing to teach and develop it. One of my son’s holistic Waldorf friends in high school referred to herself as a Waldork and unprepared in so many ways.

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      • Steve —
        I don’t disagree. I think this is, as E.D. Hirsch has eloquently explained, a Romantic hope by those in education schools that students can learn naturally, without drill.
        Cognitive science is emphatic that, to learn math and science efficiently, considerable initial drill of math and science fundamentals is cognitively essential.
        BUT if this is a war of opinion or philosophy, what sounds appealing will win.
        I believe a more effective argument is that their position is denial of science. That is true. And even among the fuzzy, cargo-cult, magical learning, pixie dust, pseudoscience, snake oil, voodoo cognition. group think citation cartel eduquacks, clear denial of science, given its fellow travelers, may be difficult to defend. — rick

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      • However, even if they decide to teach just the standard algorithms, they still have the underlying problem of low expectations and no enforced mastery at any one point in time. That’s the classic Everyday Math line – “Trust the Spiral” which turns into repeated partial learning. Our schools probably point to my son as their poster boy for Everyday Math, but nobody asked us parents what we had to do at home EVEN for our “math brain” son.

        Educators used to talk about promoting STEM in the early grades, but Common Core fails the test to give proper feedback to students and parents as to whether they are on a proper Pre-Algebra class in 7th grade track. I think the simplest solution is to pressure all schools to publicize that 7th grade math sorting requirement and to give proper feedback every year. Exceeding Common Core expectations is not enough.

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    • Eric — A personal thanks for your article from 2017. Based on it and research/articles by Kirschner, Willingham, Barry and others, several years ago I attempted to convey to my public school district the error in their decision to adopt ‘Everyday Mathematics’ (a wretched constructivist curriculum) as our CCSS-M elementary math “textbook” (textbook is in quotes because Everyday Mathematics actually does not have a physical textbook for students.) The math reformers at the NCTM that ruined math for generations of American kids after their push for constructivist curricula starting in 1989, and the architects of CCSS-M almost twenty years later, namely Phil Daro and Jason Zimba, asserted that students could achieve “conceptual understanding” without mastering basic facts and computational fluency first. Your paper helped crystallize in mind why what they were asserting was cognitively impossible. The limitations of working memory and the crucial role of long-term memory provided a cogent scientific explanation of the fatal flaw in their pedagogical theories.

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      • Sugi –
        Each year, U.S. school districts spend literally hundreds of millions of dollars of limited education funding on methods of teaching math that, like Phil Daro and Jo Boaler’s ideas, sound wonderfully appealing, but cognitive experts have proven the student brain measurably cannot manage.
        As best I can tell, the new proposed California math standards seek to return to the Phil Daro math standards of the late 1980’s and early 1990’s — a time when California fell to 49th out of the 50 states in the math NAEP.
        I very much appreciate what the parents in the LCUSD (Flintridge) district have been trying to do for 25 years to support effective math instruction.
        Perhaps a coalition might be organized to make a simple request to the California legislature – that the proposed California math standards be reviewed by cognitive experts from the state’s universities, rather than just math education professors from the ed schools?
        Such a review very well might avoid another major disaster for yet another generation of California’s children.
        — rick nelson

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      • I would be careful here. Dan Willingham who is a cognitive scientist, is prone to progressivist type thinking. In a recent op-ed, he said “The difficulty of creating “just right” hints may explain why, even though teachers say kids learn best by solving problems, in classrooms teachers often choose to explain.”

        It is as if choosing to explain is bad. Has he heard of explaining a worked example?

        The op-ed is here: https://www.latimes.com/opinion/story/2021-02-28/can-opener-problem-twitter-roderick-parent-child?fbclid=IwAR02Psd5of62PYxSHTJxbpsjGX14UHSgestewzGfYGJOJSsPUHY6yWmwK_E

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      • Eric – I do not believe the California math standards are up for revision any time soon. However, the California Department of Education (CDE) is revising the California Mathematics Framework from 2015, which was the implementation guide for the then-new California Common Core Mathematics Standards (i.e. CCSS-M) adopted in 2013. In fact, the CDE is accepting public comment on their new draft Mathematics Framework until Apr. 08, 2021. I’ve read through the first two chapters of the draft framework and they are deeply concerning. I suggest you take a look. It would take pages to explain all the problems I’ve found so far.

        The focus of the re-write of the 2015 Framework is squarely on more progressive teaching nostrums, primarily “teaching for equity.” Also included are ample servings of Boaler (e.g. the brain grows more when making mistakes!), Dweck (the power of positive thinking a.k.a. “growth versus fixed mindset”) and studies showing math is quasi-racist. In the crosshairs are tracking, acceleration, gifted programs, the “mathematics pathway system” (i.e. creating different pathways in middle and high school for students of different math aptitude and goals), Algebra 2, Calculus, “rote procedures”, accuracy, direct instruction, and anything that results in disparate outcomes by race, gender or other protected classes. If you don’t believe my characterization, here is their justification for focusing on issues of inequity in mathematics learning, quoted from the introductory chapter:

        “A fundamental aim of this framework is to respond (sic) issues of inequity in mathematics learning; equity influences all aspects of this document. Some overarching principles that guide work towards equity in mathematics include the following:
        ● Access to an engaging and humanizing education—a socio-cultural, human endeavor—is a universal right, central among civil rights.
        ● All students deserve powerful mathematics; we reject ideas of natural gifts and talents (Cimpian et al, 2015; Boaler, 2019) and the “cult of the genius” (Ellenberg, 2015).
        ● The belief that “I treat everyone the same” is insufficient: Active efforts in mathematics teaching are required in order to counter the cultural forces that have led to and continue to perpetuate current inequities (Langer-Osuna, 2011).
        ● Student engagement must be a design goal of mathematics curriculum design, co-equal with content goals.
        ● Mathematics pathways must open mathematics to all students, eliminating option-limiting tracking.
        ● Students’ cultural backgrounds, experiences, and language are resources for learning mathematics (González, Moll, & Amanti, 2006; Turner & Celedón-Pattichis, 2011; Moschkovich, 2013).
        ● All students, regardless of background, language of origin, differences, or foundational knowledge are capable and deserving of depth of understanding and engagement in rich mathematics tasks”

        You can review the California Draft Mathematics Framework for yourself here:

        https://www.cde.ca.gov/ci/ma/cf/

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    • Are they still doing that with no exceptions?
      What is their curriculum path to AP Calculus? Double up? Summer classes?
      Do they prevent incoming freshmen from taking the proper Geometry class?
      Do they just leave it up to parents and tutors to bridge that gap?
      If so, how do they rationalize that?
      How many of the unsupported students get into STEM degree programs in college?

      It doesn’t take much research to answer those questions.

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    • I found it. They created a Junior year class called:

      CCSS Algebra 2 + Precalculus Compression to lead to AP Calculus as a senior.

      If you take just Algebra 2 then you can then take regular old Precalculus as a senior.

      Really!?! My math brain son didn’t just twiddle one thumb during separate years of Algebra 2 and Precalculus that they now combine into one.

      Magic fairy dust.

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    • Barry — In the opinion piece you cite by Willingham, he writes:
      “If she wants to learn something — for example, how to open a can — for utility’s sake, not out of curiosity, an explanation makes sense…..
      But if she wants to learn because she’s curious, give her space.”
      I think the cognitive research supports this reasonable and balanced approach. And I suspect that as an instructor, Willingham realizes in schools, the parents who pay our salaries expect us to teach their children “for utilities sake:” so they can earn a living in a technological economy, whether they are expressing “curiosity” at the moment about the skills they will need or not. — rick

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  4. No time to dig into details – but there is a lot of ignorance of math research across various domains here. Please hold your opinions people and study a little more.

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    • Ramani –
      Many of the people doing “math research” are making claims that children and young adults can reason in ways cognitive scientists say their brains cannot do. The way the brain works is the way cognitive experts agree the brain works.
      How much do you know about the strengths and weaknesses of working memory — where the human brain solves problems? Please answer.
      Do you not aware of what scientists say about how the brain learns and how it solves problems? Or are you denying their consensus science? Please answer.
      If you are going to make accusations of ignorance, a serious charge, shouldn’t you make some effort to be specific and substantiate your claims?
      Barry has written his views. I believe the cognitive science summarized at http://www.ChemReview.Net/CCMS.PDF offer support for his views. Where specifically do you disagree?
      — rick nelson

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    • “… there is a lot of ignorance of math research across various domains here. Please hold your opinions people and study a little more.”

      “Ignorance”??? You’re guessing.

      “Opinions”???? Maybe that’s all YOU have.

      “Research”???? How about reality and results?

      Many of us have lots of experience in reality and results – teaching, real STEM degrees, AND real STEM work over decades.

      I’ve taught and tutored math and I watched and helped my son survive “Trust The Spiral” EM and CCSS over the last two decades. I have 50 years developing mathematical software. If you have any new “research,” post links here. I really doubt we haven’t seen it before. Better yet, point to real life cases of success. This fuzziness has been going on for decades.

      Where’s the beef?

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    • Is this the kind of research you’re talking about?

      “2020 – 2023 NewSchools Venture Fund, EF+Math Program, Developing Low-cost Mobile App Technology to Assess Ability and Fluctuations in Executive Functions and Math Learning. G. Ramani (PI); Subcontract with Washington University in St. Louis, D. Barbour (PI)”

      Are you assessing “executive functions”, developing “low-cost mobile app technology,” or both? Does the assessment include finding out whether kids have mastered their adds and subtracts to 20 or the times table? Do you need a mobile app to do that or does that just help you get funding?

      What do you do if they haven’t mastered these basic skills? What is the feedback correction loop? What grade level does it happen at? Are kids allowed to move on to new grade/material if they haven’t mastered the old material? Why? That doesn’t happen in high school or college, so why K-6?

      These feedback and enforcement loops have been gone from K-6 for at least 2 decades. My son was in full inclusion K-8 schools where they “trusted the spiral.” It didn’t work and my son and all of his STEM-prepared friends had to get help at home and with tutors. We parents enforced mastery of the basics and ensured that our kids were properly prepared beyond the NON-STEM Common Core level. Common Core INCREASES the academic gap. Math is taught at a single slope from Kindergarten to a senior in high school who is likely to pass a “College Algebra” course. It requires parents and tutors to get their kids to change that slope.

      That would be simple research. Ask the parents of all of the STEM-degree ready students what we had to do at home or with tutors. There is an existing STEM pathway that works. If you don’t like it, then you are free to try to create a new one. Just DO NOT force it down the throats of all students and parents in K-8. We STEM parents are tired of CYA.

      If you want to see what works for under-privileged groups – as equals, see El Sistema for music. They take kids from the barrios and get them to play at Carnegie Hall and the BBC Proms by the end of high school. The key to it are the private lessons that enforce mastery of skills and start in the earliest grades and NOT the mixed-ability orchestra classes in school. These musicians do not have just mere facts or rote skills. Mastery drives understanding and excitement and hard work – NOT they other way around.

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