The Pedagogical Agenda of Common Core Math Standards

by Barry Garelick Mathematics education in the United States is at a pivotal moment. At this time, forty-five states and the District of Columbia have adopted the Common Core standards, a set of uniform benchmarks for math and reading. Thirty-two states and the district have been granted waivers from important parts of the Bush-era No [...]

by Barry Garelick

Mathematics education in the United States is at a pivotal moment. At this time, forty-five states and the District of Columbia have adopted the Common Core standards, a set of uniform benchmarks for math and reading. Thirty-two states and the district have been granted waivers from important parts of the Bush-era No Child Left Behind law. As part of the agreement in being granted a waiver, those states have agreed to implement Common Core. States have been led to believe that adoption of such standards will improve mathematics and English-language education in our public schools.

My fear (as well as that of many of my colleagues) is that implementation of the Common Core math standards may actually make things worse. The final math standards released in June, 2010 appear to some as if they are thorough and rigorous. Although they have the “look and feel” of math standards, their adoption in my opinion will not only continue the status quo in this country, but will be a mandate for reform math — a method of teaching math that eschews memorization, favors group work and student-centered learning, puts the teacher in the role of “guide” rather than “teacher” and insists on students being able to explain the reasons why procedures and methods work for procedures and methods that they may not be able to perform.

I base my opinion on what I see being discussed at seminars on how to implement the Common Core. The emphasis in such forums is not on the content standards, but on the 8 Standards of Mathematical Practice (SMP). The SMP are a slight repackaging of the National Council of Teachers of Mathematics (NCTM’s) process standards. And while some maintain that “process” doesn’t mean the same as “practice”, from what I see, process is still trumping content. The popular interpretation of SMP is a pedagogical agenda that features student-centered and inquiry-based approaches. The practice of “making sense of mathematics” sounds great on paper. But what it means to those of the thoughtworld of the education establishment is what is also called “habits of mind” in which students are taught habits of analyzing problems long before they have learned the procedural knowledge and content that allows such habits to develop naturally. They are called upon to think critically before acquiring the analytic tools with which to do so. More precisely, they supposedly are acquiring the analytic tools by being given problems to solve and learning via their groups and exploration (with teachers “facilitating”) and being forced to learn the techniques in a “just in time” basis. Such a process while eliminating what the edu-establishment views as tedious “drill and kill” exercises, results in poor learning and lack of mastery.

In addition, both the SMP as well as the content standards themselves are predicated on a belief that conceptual understanding MUST precede procedure. Evidence that this belief will have widespread implementation is seen in a recently published document that provides guidance to publishers on criteria for aligning textbooks to the standards. ( See http://www.corestandards.org/assets/Math_Publishers_Criteria_K-8_Summer%202012_FINAL.pdf ) Two of the writers of this guidance document –Phil Daro and William MacCallum–are the lead authors of the math standards. The document states that “conceptual understanding needs to underpin fluency work,” or that “[sufficient] fluency can be practiced in the context of applications.” (Found on page 11 of the referenced document.) It is untrue that conceptual understanding “needs” (implying it always does) to underpin fluency. Often it does, often it does not..Understanding and procedure work hand in hand; sometimes students learn procedure before understanding the concept.

While the math standards may be an improvement over existing standards in some states, they are still largely deficient. Members of the U.S. Coalition for World Class Math have addressed the content standards in comments submitted to CCSSO and NGA. These comments are on the web site of the U.S. Coalition for World Class Math: (http://usworldclassmath.webs.com/U.S.%20Coalition%20for%20World%20Class%20Math%20Comments%20on%20June%202010%20CCSSI%20Math%20Standards.pdf)

Ze’ev Wurman has also written extensively about these standards in a report published by the Pioneer Institute. (See http://www.pioneerinstitute.org/pdf/common_core_standards.pdf ) He is an executive in the high tech industry in Silicon Valley and was a member of the 2010 California Academic Content Standards Commission that evaluated the suitability of Common Core’s standards for California. He served as a Senior Policy Adviser with the Office of Planning, Evaluation, and Policy Development at the U.S. Department of Education from 2007 to 2009. I echo his concerns with the content standards as summarized below:

– Common Core replaces the traditional foundations of Euclidean geometry with an experimental approach. This approach has never been successfully used in any sizable system; in fact, it failed even in the school for gifted and talented students in Moscow, where it was originally invented. Yet Common Core effectively imposes this experimental approach on the entire country, without any piloting.

– Common Core excludes certain Algebra II and Geometry content that is currently a prerequisite at almost every four-year state college. This effectively redefines “college-readiness” to mean readiness for a nonselective community college, as a member of the Common Core writing team acknowledged in his testimony before the Massachusetts Board of Elementary and Secondary Education.

– Common Core fails to teach prime factorization and consequently does not include teaching about least common denominators or greatest common factors.

– Common Core fails to include conversions among fractions, decimals, and percents, identified as a key skill by the National Research Council, the National Council of Teachers of Mathematics, and the presidential National Advisory Mathematics Panel.

– Common Core de-emphasizes algebraic manipulation, which is a prerequisite for advanced mathematics, and instead effectively redefines algebra as “functional algebra”, which does not prepare students for STEM careers.

More specifically, at the K-8 grade span:

– Common Core does not require proficiency with addition and subtraction until grade 4, a grade behind the expectations of the high-performing states and our international competitors.

– Common Core does not require proficiency with multiplication using the standard algorithm (step-by-step procedure for calculations) until grade 5, a grade behind the expectations of the high-performing states and our international competitors.

– Common Core does not require proficiency with division using the standard algorithm until grade 6, a grade behind the expectations of the high-performing states and our international competitors.

– Common Core starts teaching decimals only in grade 4, about two years behind the more rigorous state standards, and fails to use money as a natural introduction to this concept.

– Common Core fails to teach in K-8 about key geometrical concepts such as the area of a triangle, sum of angles in a triangle, isosceles and equilateral triangles, or constructions with a straightedge and compass that good state standards include.

There is already evidence that book publishers’ revisions to texts that align with the standards are highly likely to be “inquiry-based”. (See http://www.virtual-strategy.com/2012/08/22/houghton-mifflin-harcourt-offers-math-expressions-common-core-2013-support-national-scien Of note is this statement : “With special emphasis on Mathematical Practices and Mathematical Progressions, teachers create an inquiry-based environment and encourage constructive discussion.”) Discovery and group learning approaches to math have had poor results when they have been used in classrooms across the country. The Common Core math standards will in effect be a national mandate for reform math. I do not believe they will be good for this country.

Barry Garelick has written extensively about math education in various publications including Education Next, Educational Leadership, and Education News. He recently retired and has obtained his credential to teach math (middle school/high school) in California.

Comments

SteveH

September 28, 2012 at 6:06 am

CCSS doesn’t solve K-6 math problems. It institutionalizes them. Talk of fluency in the standard is extraordinarily limited, and the tests will reflect that. Schools can, and will, continue to claim the higher ground of understanding while bright students continue to struggle with basic skills. Schools will continue to use “trust the spiral” curricula like Everyday Math and assume that students who don’t do well must have cognitive issues. Many think that the spiral works, because, well, it just does. K-6 schools want to be math “pumps” rather than math “filters”, but they are really delaying the filter until there is nothing students and parents can do about it. By then, the problem will look like it belongs to the kids, the parents, peers, and society, and the kids will believe it.

Knowledgeable parents will still have to ensure mastery of the basics in K-6 and will still have to guide their kids through the nonlinear transition between the low-expectation, process-oriented, problem solving math fairlyland of K-6 to the rigorous door-opening, skill-based STEM curriculum of high school. Many will continue to see a correlation with affluence and IQ, but fail to investigate what cognitive level of help was provided each step of the way to the best students. When schools send home letters to parents telling them to practice “math facts” with their kids (I’ve received many.), affluence reflects the time, money, and willingness to do the school’s job, not the cognitive level of the material.

CCSS does not fix the basic mastery issue of K-6. Schools will continue to talk about critical thinking and understanding while bright kids continue to get to fifth grade with very bad basic math skills. CCSS has it backwards. It is too weak on K-8 math, and then sets the bar too high in high school for many. Schools will continue to fail to ensure mastery of the basics in the lower grades, but then torture kids to reach a pseudo algebra II level in high school without regard to what their career objectives might be. The CCSS workplace analysis lumped everyone into the same career math boat. It’s too weak for a STEM career and not appropriate for many of the rest. The CCSS math standard defines a mountain top for math, not a base camp for a STEM degree. CCSS basks in the glow of a statistically high bar in high school, but does nothing about ensuring that kids have any better chance of getting there from the lower grades.

Reply
BobDean

September 28, 2012 at 8:17 am

The document states that “conceptual understanding needs to underpin fluency work,” or that “[sufficient] fluency can be practiced in the context of applications.” It is untrue that conceptual understanding “needs” (implying it always does) to underpin fluency. Often it does, often it does not.

I would go even further Barry. In reality, math is a tool used to accomplish other goals. Knowing how to use it is much more important than knowing why it works. In fact, it is not necessary at all to understand why in order to use the tool of mathematics. Do you have to understand why fuel injection works before you can push on the gas pedal to make you vehicle move?

In the construction world where I spent 17 years math is used on a daily basis. The workers don’t have to understand the Pythagorean Theorem in order to understand that if the diagonals are equal the foundation is square. Nor do they care why…. they just square up the foundation and move on. That is how most mathematics is used in the real world.

I learned mathematics as a tool long before I understood why much of it worked. The misguided thinking in the CCSS will continue to leave students mathematically illiterate. It is not as important to understand that multiplication is repeated addition as it is to know your multiplication tables.

The CCSS and the emphasis on the SMP is a very sad day for education in this country. In my opinion, the best way out is school choice for parents and students.

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Education Week: How Common Core Math Dumbs Down Students « COMMON CORE

September 28, 2012 at 9:20 am

[...] article here: http://www.educationnews.org/education-policy-and-politics/the-pedagogical-agenda-of-common-core-mat... Share this: Pin ItEmailPrintShare on TumblrDiggLike this:LikeBe the first to like [...]

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teacher losing patience

September 30, 2012 at 7:50 pm

I wonder if it is possible to bring a class action lawsuit on behalf of our children and teachers who are subjected to this insanity. Our nation’s future is in jeopardy with this hogwash of math that the CCC is. I am originally from one of the countries that beat US on every math and science tests. I have been teaching here for almost ten years. I hear a lot of rhetoric and hype about improvements being made in our education, new ways of teaching, technology, etc. All of it is rubbish. This is not a world class education, but a world class deception. Being part of the system and on the payroll of the system I have to jump though all the hoops, but I still teach my way and my students do better than anybody else’s. Still I am far from being satisfied. Moreover, I started to feel hopeless. What I teach in 9th grade in US would be taught in 4th grade in my country of origin. How did we come to the point that children can’t add or subtract in 9th grade? What critical thinking can we talk about if algebra honors students do not know the difference between the area and perimeter, or even that triangle has three sides! My students are not coming from low socio-economic backgrounds, so what excuses can I make for them? Why do they seem to be unable to remember anything we teach, no matter what methods we use? What I experience in my classroom is similar to what Bill Murray’s character was going through in the Ground Hog Day movie. How do I successfully teach algebra if they lack basic skills such as writing legibly their numbers and variables so they can read their own handwriting? Or, God forbid, adding two fractions. With my educational background, I am capable of preparing my students for STEM careers, international competitions, yet I only have them for one year, and every year I have to start with teaching that triangles have three sides, and that adding two fractions with unlike denominators requires a common denominator. So much for my ambitions as well as my students’. CCC is just the last nail in the coffin of the US math education.

Reply
- Barry Garelick
  
  October 1, 2012 at 10:29 am
  
  Teacher Losing Patience: Thank you for your comment. Regarding the legality of Common Core, there are some who believe that federal funding tied to Common Core is unconstitutional. For more info about this, visit the Truth in American Education website at http://truthinamericaneducation.com/
  
  Reply
Barry Garelick: Common Core Math Standards, A Mandate for Reform Math | Truth in American Education

October 3, 2012 at 12:14 pm

[...] Reform Math Posted on October 3, 2012 by Shane Vander Hart Barry Garelick, a TAE advocate, wrote an op/ed for Education News. Here’s an excerpt: Mathematics education in the United States is at a pivotal moment. At [...]

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Shelley

October 3, 2012 at 7:53 pm

The CCSS require that students “By end of Grade 2, know from memory all sums of two one-digit numbers.”

Maybe you’re referencing 4.NBT.4 – “Fluently add and subtract multi-digit whole numbers using the standard algorithm.” If that’s so, I understand that it would be a concern that while requiring the standard algorithm in name, delaying it’s mention until the 4th grade would enable the reform approach (rather than standard algorithm) to addition and subtraction to be firmly implanted.

My assumption would be that Everyday Math’s route would be common for reform math programs and I know that in their CCSS edition they teach the standard algorithm in the 2nd grade (this is a change from prior editions). MY Math (kind of a hybrid reform) teaches it in 2nd grade as well. Both apply the standard algorithm to numbers up to 3-digits. They continue to teach alternative algorithms as well. Do you know of any reform programs that are approaching it differently?

Reply
Shelley

October 3, 2012 at 7:57 pm

The CCSS edition requires “By end of
Grade 2, know from memory all sums of two one-digit numbers.” (2.OA.2)

Maybe you’re referencing 4.NBT.4 – “Fluently add and subtract multi-digit whole numbers using the standard algorithm”? If so, I understand that a concern could be that while requiring the standard algorithm in name, delaying it’s mention until the 4th grade enables the reform approach to addition and subtraction to be firmly implanted.

My assumption would be that Everyday Mathematic’s route would be common for reform math programs and I know that in their CCSS edition they teach the standard algorithm in the 2nd grade (this is a change from prior editions). MY Math (kind of a hybrid reform) teaches it in 2nd grade as well. Both apply the standard algorithm to numbers up to 3-digits. They continue to teach alternative algorithms as well. Do you know of any reform programs that are approaching it differently?

Reply
- Barry Garelick
  
  October 4, 2012 at 6:22 am
  
  Investigations in Number, Data and Space delays teaching the standard algorithm for addition and subtraction until 4th grade. The fact that other books may present it in second grade does not provide assurance that other publishers will do the same. If the standards allow it to be presented in the 4th grade, then what’s to stop publishers from doing that?
  
  Reply
Mike Kelley

October 5, 2012 at 10:35 am

“– Common Core does not require proficiency with addition and subtraction until grade 4, a grade behind the expectations of the high-performing states and our international competitors.”

This statement is simply not true. Standard 3.NBT.2 requires students to fluently add and subtract three digit numbers in Grade 3. Not just proficiency, mind you, fluency. Students are graduating from addition and subtraction to multiplication and division in Grade 3. Of the 24-ish standards (not counting sub-standards), only ONE refers to addition and subtraction. If you conclude that students have not yet mastered the algorithm by Grade 3, I am simply not reading the same standards you are.

This is not the only such misrepresentation in this article. I appreciate that you may not agree with the approach, but it is another thing altogether to misrepresent the standards. Further, there is much moaning and gnashing of teeth that a new technique flies in the face of “the way we do things.” We as a nation are in no position to boast about the mathematical achievement of our students, so to criticize a new approach by comparing it to the status quo and thereby find it lacking is a flawed argument.

You argue, often misleadingly as I have demonstrated, that the CCSS are a reduction in the rigor of current standards. Even if that were true (which I argue it is not), rigorous standards do not successful students make. Metaphorically speaking, I can set the most rigorous weight loss standards in the world but if they do not translate into results or actually work in concert with my body chemistry, then there are merely words and help no one.

I also find it strange that so many are up in arms against conceptual understanding. It’s not as though conceptual understanding and solving problems are either-or. No one is arguing that we languish in concepts and ignore computational fluency–that sort of argument is naught but a straw man.

The most legitimate concern, which is not adequately addressed here, is teacher training. While so many spend time raising their fists to the sky about policy, we’d be better spent ensuring teachers are equipped to implement this mathematical strategy, one which I have seen in action and find absolutely groundbreaking if implemented well.

My worry for math education is as it has always been with politics in general–that policymakers spend too much time arguing for or against strategies and not enough time equipping our educators, the best of whom are eager to implement any strategy that could potentially foster deeper and more elastic learning.

Reply
Barry Garelick

October 5, 2012 at 12:25 pm

Mike:

You say “Standard 3.NBT.2 requires students to fluently add and subtract three digit numbers in Grade 3.”

Here’s what the standard says: “Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.”

In 4th grade, standard 4.NBT4 says: “Fluently add and subtract multi-digit whole numbers using the standard algorithm.”

My point is that the teaching/learning of the standard algorithm for addition and subtraction is delayed until 4th grade. In the interim, students use “strategies and algorithms based on place values…” etc to do this. I.e., not the “standard algorithm”. This manifests itself as the approach that has been used in Everyday Math, and Investigations in Number, Data and Space, in which the algorithms students use in the beginning are inefficient. They are also called “student-invened algorithms”, but this is misleading. They are not invented by students but prescribed by these programs.

The draft CC math standards required mastery of the standard algorithm for addition and subtraction in the 2nd grade. Why the delay? I am assuming it’s because by deferring it, they believe students are gaining an “understanding” of the mechanism before they are taught the efficient procedure.

You state: “No one is arguing that we languish in concepts and ignore computational fluency–that sort of argument is naught but a straw man.” I beg to differ. The guidance for publishers of math books that I cited in the article states “conceptual understanding needs to underpin fluency work”. It doesn’t always need to underpin procedural fluence. Often, procedural fluency can come first. To say that “no one is arguing” this is a misrepresentation. I hear the argument that understanding should come first before procedural fluency in many presentations and meetings on math education. In fact, this argument is going on in the provinces of western Canada, where the education ministry is insisting that understanding must come first. Sometimes understanding comes first; sometimes procedural fluency. Consider how you learned how to count. You pointed to an object and gave it a number. Did someone first make you understand that counting is a one-to-one correspondence of elements of one set with a subset of natural numbers and the highest element in the natural number subset is called the “cardinality”? I doubt that they did. I imagine you learned how to count first without knowing the conceptual underpinning.

Reply
SteveH

October 6, 2012 at 7:24 am

The use of the words fluency and fluently in the standard is a joke.

“No one is arguing that we languish in concepts and ignore computational fluency–that sort of argument is naught but a straw man.”

“Ignore”? But nobody is complaining that people say that. What many of us are complaining about is that computational fluency is not ensured and does not cover enough topics. Count the times “fluently” is used in the standard. When my son was in fifth grade at an Everyday Math school, the school had a meeting with the parents about math. They all talked about balance, but nobody defined or calibrated it. Of course, everyone love balance. However, the school had to form an extra help class for bright students who were still struggling with the times table. But it’s not just the times table. What about fluency in fractions, percents, decimals, signed number calculations, and on and on? The CCSS standard just throws in a few token fluency requirements.

The only thing that will define fluency in the standard are the tests that the states will use. Ours will use PARCC, and we await examples of the test and what the results will mean. I don’t expect much considering that less than 50% of our high school students meet the proficiency standards of our current NCLB test.

While it is true that national standards do not guarantee effective teaching and curricula, they can push schools to pay more attention to the skills side of the balance equation. However, it will have a difficult time changing what’s in the hearts and minds of K-8 educators. They may not ignore fluency, but they surely devalue it.

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SteveH

October 7, 2012 at 5:56 am

“I also find it strange that so many are up in arms against conceptual understanding. It’s not as though conceptual understanding and solving problems are either-or.”

Nobody argues against conceptual understanding, but it better evolve into something more than “conceptual”. There are many different levels of understanding, but what are the different levels and how do you get there, from the top down or the bottom up. What is the process of going from a conceptual understanding to one that is flexible?

When people talk about understanding things like multiplication with multi-digit numbers, are they talking about a place value understanding, an algebraic understanding, or a base ‘n’ understanding? Do we expect students to be able to design their own “lattice method” for base 10 or base ‘n’? How about being able to explain how the standard algorithm is more efficient than the Partial Products algorithm?

It’s nice to have a conceptual understanding and be motivated by real world hands-on problems, but where do you go from there. What, exactly, is “deeper and more elastic learning”? You can’t just argue generalities. There is linkage between mastery of basic skills and understanding. Nobody can survive long in math with some vague rote knowledge or some vague conceptual understanding. Whether you attack the goal of mastery and understanding from the bottom up or the top down, you have to finish the job. You have to define and know what the job is. You cannot assume that showing rote skills on badly-defined tests is good enough, and you can’t assume that happy, active learning kids in a conceptual learning environment will magically have the engagement and motivation to properly master skills. Educators see rote skills and assume that all they need is to do is approach the problem from the top down. They feel better about all of the active learning they see, but they get no further to the real goal. Actually, they don’t get as far.

Unfortunately, most K-6 schools want to take the engagement and conceptual top-down hands-on approach and then “trust the spiral” with curricula like Everyday Math. It doesn’t work. Schools are not ensuring mastery of basic skills. There is too much weasel room in the CCSS standard to get them to change. They want math to be a pump and not a filter, so they set low expectations, talk about the wonders of critical thinking and problem solving, pump the kids along, and hope for the best. By the time anyone figures out there is a problem, it’s too late. They see some kids who do well, but they don’t bother to ask their parents what they do at home. Some begin to think that the problem for the others is really “body chemistry” or IQ.

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Kristin

October 13, 2012 at 4:33 pm

I completely agree with this article. I am someone who is very mathematical. I will admit that I did not fully understand why the traditional algorithm worked until I was in middle school, but I could do it in elementary school. I am not sure when my brain fully got it, I just know that a lot of math I knew how to do, but did not completely get it until a later date. I am teaching the traditional algorithm this year to my third graders, but was told next year with CC I will not be allowed to. WHAT! That they should use mental math, and other strategies to add. Crazy! I am so outraged that I have decided my child is NOT going to public schools until CC falls flat. Which it will, considering the teaching techniques it requires sound nice, but are not based in reality. It reminds me of the pod concept of classrooms with no walls in the 1970s. Sounded great, teachers could team up and work together, students would be able to go from class to class if needed. In reality it was a loud, distracting mess. Most schools built like this have since been remodeled so that they are not pods. My child is going to go to a private school were fact mastery and the traditional algorithm are still essential components. A school were science and social studies are still taught, as are the literature that is fiction. It is my belief that if we still taught science and social studies like we use to there would be no need to include more nonfiction material in our reading program, the science and social studies is the nonfiction.

Reply
- Barry Garelick
  
  October 14, 2012 at 5:17 pm
  
  Kristin,
  
  Thanks for your comment. I find it interesting that you are being told you are not allowed to teach a traditional algorithm in your school. Are you told this despite it being included in the textbook you are currently using?
  
  You may be interested in an article I just wrote for online Atlantic on math teaching. It is located at:
  http://www.theatlantic.com/national/archive/2012/10/its-not-just-writing-math-needs-a-revolution-too/263545/
  
  Reply
- Elizabeth
  
  October 18, 2012 at 9:03 pm
  
  They implemented CC this year in our school system in TN. I have a third grader who loved math and got A’s in math until this year where he struggles to get a C. He struggles with “explaining” how he got his answer after using “mental math”. In fact, I had no idea how to explain it! It’s math 2+2=4. I can’t explain it, it just is. I do not know what to do, and have considered pulling him out of public school and homeschooling him. What home schooling curriculum would you recommend? Since this is now mandated I don’t see any other option.
  
  Reply
  - Barry Garelick
    
    October 21, 2012 at 2:31 pm
    
    Saxon Math has good homeschooling texts, as does Sadlier Oxford. Singapore’s Primary Math (US Edition) series is also very good. Check out the Singapore stuff at http://www.singaporemath.com. Sorry to hear they are implementing Common Core in this way.
    
    Reply
COMMON CORE

October 14, 2012 at 4:20 pm

[...] http://www.educationnews.org/education-policy-and-politics/the-peda… [...]

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Ammon

October 17, 2012 at 2:18 pm

This has been an interesting topic to research for me. I appreciate hearing the different perspectives on these issues.

I teach an elementary math methods course for pre-service teachers and this debate is exactly what I am trying to help my students dig into carefully. We are going to have a debate in our next class specifically looking at the benefits of teaching for understanding vs. the benefits of practicing/focusing on basic skills.

I don’t pretend to have a perfect answer to this, but I wanted to share a few of my thoughts.

I think that one aspect of this issue that the author of the article hasn’t addressed is the motivational aspects of the different epistomologies. One of the main features of a traditional math program that focuses on basic facts is the idea that if you do enough practice, the students will eventually get those basic facts down. Assigning a large amount of practice for basic skills is essentially built on the cognitive model that when something is done correctly often enough then it becomes encoded in our brains. This has worked for many students and I don’t think should be abandoned entirely.
The problem with this focus to me is that in my own experiences in math, my teaching of 4th graders for 6 years and most recently my 2nd grade daughter is that too many students end up dreading math. Motivation to continue with math in any way becomes pretty rare except for the most successful students.

Teaching for understanding and critical thinking also assumes that information is being encoded and that practice is needed, but the focus is simply on encoding (thinking about) the deeper understandings of why the algorithim works. If done poorly in class, this can be pretty useless, as many have argued, but when it is taught well, I have found that more of my students are motivated to work with math problems. My biggest concern with this method is the time that it takes to really let students think about and discuss math at a deep level. (One way to help with this is to have students explain their thinking in pairs or small groups while the teacher circulates to assess understanding)

Overall, my view as a former elementary teacher and current professor is that creating a balance between these issues really would be the best math learning environment. The benefit of balance would be most likely when we reduce (but not eliminate) the drilling of basic facts and also provide opportunities in class and in homework for students to explain their thinking thus building fluency and understanding. Not easy or simple to do but I think it is possible.

Reply
Barry Garelick

October 17, 2012 at 3:29 pm

Thank you for your comment. The traditional model of math education is often misrepresented as too much drill and tedium. Like any method, the traditional one can be done poorly. There are ways to achieve the necessary practice without overdoing it.

But more importantly is what is meant by “teaching for understanding”.

There are natural progressions of learning throughout the elementary school curriculum. In some cases, understanding comes first. In others it comes much later. Artificially imposing the sequence as in “understanding MUST come first” will sometimes work. But it is unhealthy and unbalanced.

Consider how a child learns to count. There are two aspects: (i) the mechanical act of pointing at objects and reciting the number sequence: one, two, three, … and (ii) the abstract conception of cardinality and cardinal equivalence as established by one-to-one correspondence, and of course eventually (iii) the marriage of (i) and (ii) into a coherent and fundamental concept of “number”.

Now, (i) is simply a skill. It is learned largely by rote. (ii) is, arguably, the “understanding” element. It is learned by reflection and analytic interaction with ideas. Which comes first? Perhaps there are exceptions, but I have NEVER seen a child who fully grasps the number-concept “understanding” embodied in (ii) prior to learning the fundamental counting skill (i). This is a clear instance in which skill precedes understanding.

More to the point, it is the skill itself upon which understanding is built, in this case. The child develops his understanding (ii) by repeatedly practicing the pure skill (i) over and over until it is realized conceptually through familiarity and tactile experience that forges pathways and connections in the brain.

Daniel Ansari, a leading scholar of cognitive developmental psychology who studies brain activity during the learning of mathematics said, at a recent conference, that neither skill nor understanding should be underemphasized — they provide mutual scaffolding and both are essential. But in terms of sequential priority, there is no chicken-and-egg problem: more often than not, skill must come first, because it is difficult to develop understanding in a vacuum, whereas skill can be developed in this fashion, and provides the appropriate experiential context within which understanding can be developed.

Reply
- Ammon
  
  October 18, 2012 at 12:04 pm
  
  Thank you for your extended comments, I appreciate the depth of discussion, they help to create in my own understanding.
  
  Reply
SteveH

October 18, 2012 at 5:58 am

Mastering skills is not just rote. There is linkage to understanding at every grade level and with every skill. This is more clearly seen when you get past arithmetic and into fractions. Many educators want to deny this. They think it’s like memorizing the list of presidents. When educators talk about “drill and kill”, they try to imply that it’s more than just “kill”. They imply that the skills are really not that important, or that there is some other way to get there. This leads them to a top-down approach starting with conceptual ideas, group work, and hands-on real world problems. These fun “active learning” classroom activities take a lot of time and don’t translate into the necessary skills. Reform math has never come to grips with this problem. Parents have to ensure mastery at home, and teachers are assuaged by seeing some students who are successful.

Is there a happy and fun process to success in math? Is there a natural process to mastery? No. Students have to realize that learning may require hard work. It may require flash cards. It will require nightly homework sets in math. If kids are unhappy because learning is not fun all of the time, they are getting the wrong message. They need to know that hard work can lead to so much more fun. What’s worse is when students spend class time in fun group learning and then find themselves completely lost when it comes to applying math skills. It’s one thing to not like math, but be able to do it, but another to think you like math, but can’t do a thing.

So what does rote mean in math when you get into fractions? Can you do all sorts of problems without understanding? I’m not talking about pie chart conceptual understandings. Let’s look at the supposedly rote rule of “invert and multiply” for dividing fractions. What level of understanding is required here? Does this understanding come just from talk of concepts or by mastery of skills? How formal do you want to get? Reform math sticks with conceptual understandings that leave kids high and dry when it comes to solving all of the variations one might encounter. Neither rote learning nor general concepts will get the job done. However, is rote learning all that happens with a traditional (skills first) approach to math? Is there no linkage between skills and understanding? Does success in solving problems show no understanding?

What if you have something like:

5 / (2/3)

or

(5/3) / (-3/8)

or

(2 1/2) / 4

or

3.25 / (8/3)

or

(3x/2y) / 5x

Can all of these problems be done with just rote understanding or just concepts? What process does reform math use to go from general pie chart concepts to the understanding needed to do these problems? How does reform math ensure mastery of skills? It doesn’t. It sees little linkage between understanding and mastery of skills.

What happens when students get to word problems such as DRT, work, and mixture? Maybe you can get away with a general Polya-type process, but what happens when you add geometry and angles? How do you develop the understandings required to properly apply specific mathematical skills? You can’t use Polya to solve fluid flow problems in pipes without understanding and mastery of Bernoulli’s equation. It takes a lot of practice and understanding to develop the flexible skills to apply mathematical definitions, identities, and algebraic rules to any situation. This process is glossed over in K-8 reform math education.

Motivation and engagement don’t get the job done. It’s like getting fired up by diet commercials. The next day, you are back to eating mathematical Twinkies, and educators will start looking for the next big mathematical diet. Project Lead the Way in high school will never motivate away the gaps students have in math skills. Colleges care about math, not Project Lead the Way. Engagement is a cover for putting the onus of mastery on the students.

Balance is a false god. Balance doesn’t ensure mastery of skills. Mastery of skills is the goal. Understandings can be improved, but gaps in skills are very difficult to diagnose and fix. They are also a sign of missing understanding. A top-down approach will never get the job done because educators think of skills as rote and as something that is really not quite that important. They think that understandings can drive problem solving. They cannot. You need the understandings that ONLY come from mastery of skills. Educators need to ask the parents of their best students what they do at home. We ensure mastery at every step of the process even if our kids are “math brains”.

As Barry explained:

“… there is no chicken-and-egg problem: more often than not, skill must come first…”

Bottom up, not top down. Flexible skills are never rote. Skills can be made more flexible with additional understanding and practice, but conceptual understanding without mastery of skills is nowhere and leads to a dead end.

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Common Core won’t improve education — GraniteGrok

October 26, 2012 at 5:56 pm

[...] Common Core, by the end of eighth grade students will be two years behind their international counterparts in math. High school graduates will achieve only a seventh-grade [...]

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