The Common Core’s Pedagogical Tomfoolery

One frequently hears that the Common Core standards are merely standards and expectations that do not dictate curriculum or pedagogy. Common Core proponents argue that those national standards do not interfere with the ability of teachers to use their preferred pedagogical approaches, and do not further interfere with local autonomy over the curriculum. Here, for example, are Kathleen Porter-Magee and Sol Stern making the case why conservatives should support the Common Core:

Here’s what the Common Core State Standards are: They describe what children should know and the skills that they must acquire at each grade level to stay on course toward college- or career-readiness, something that conservatives have long argued for. . . . The Common Core standards are also not a curriculum; it’s up to state and local leaders to choose aligned curricula.

Indeed, on the face of it, this is exactly what the Common Core standards claim to be. Its English Language Arts standards announce:

A focus on results rather than means

By emphasizing required achievements, the Standards leave room for teachers, curriculum developers, and states to determine how those goals should be reached and what additional topics should be addressed . . . . Teachers are thus free to provide students with whatever tools and knowledge their professional judgment and experience identify as most helpful for meeting the goals set out in the Standards. (p.4)

And they categorically state:

The Standards define what all students are expected to know and be able to do, not how teachers should teach. (p.9)

Similarly, the Common Core mathematics standards call themselves content standards – in other words, they dictate the “what” rather than the “how.”

Are these claims true? All around the country we hear of parents tearing their hair out after they look at what their children now bring home carrying the label “Common Core.” We hear stories of children providing correct answers to arithmetical problems and being marked down for using “improper procedures.” We hear about Common Core teacher training stressing the “how” rather than the correctness of students’ results. Are those anecdotes just isolated incidents and examples of wrong-headed interpretations of the standards?

If one reads the standards themselves, it quickly becomes obvious that they are not only about the “what” but rather include a lot of the how, despite their claim to the contrary. And that many of those anecdotes describe not a wrong-headed interpretation of the standards, but rather a faithful implementation of what they explicitly demand.

In English Language Arts much discussion occurred around the standards’ directive to share class reading time evenly between informational texts and literary texts. In high school, the standards insist on increasing the informational texts share to 70% (which may include also reading outside English class). I will not spend much time here on the foolish reasons for this change – that this is how the NAEP test splits its items, which has little to do with how to teach reading – but I’d simply point out that this is a curricular directive par excellence. It orders teachers how to structure their class time.

In mathematics, my own area of expertise, the examples of curriculum and pedagogy are numerous. Look, for example, on a first grade standard:

1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Were this a true content standard, it would have simply stopped after its first sentence: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Yet the standard continues and lists at least four different ways students must use to show … what? Can’t they simply show they can add and subtract, correctly and fluently?

And lest you think this is just a fluke, here is essentially the same standard in the second and third grades:

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

3.NBT.2: 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 other words, students are not allowed simply to learn how to add and subtract in first grade, in second grade, or in third grade. No, they must use the training wheels that the authors want them to use, whether they can ride without them or not. What is this if not pedagogy, and a wrongheaded one to boot? Young children do not need four different ways to “explain” addition – at best, this could be guidance to teachers how to individualize teaching rather than expect children to know all these ways.

One can argue that those are just suggestions. Unfortunately, this is incorrect. The Common Core assessment consortia (PARCC and SBAC) will test these wrong-headed “strategies,” paying attention to the variety of ways problems are answered rather than to correctness of results.

Perhaps the most egregious case of imposing pedagogy occurs in Common Core geometry. It expects the teaching of triangle congruence in a particular and experimental way:

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

A true content standard would simply say “Students prove triangle congruence” or, perhaps, “Students understand triangle congruence,” leaving the method of instruction to the teacher. Instead, Common Core not only dictates how to teach congruence, it insists on a specific experimental method of instruction that has an established a track record of failure where it was invented (pg. 33-35 here). Turns out that the authors of the standards were unaware of this record, and simply thought it mathematically “neat.” Talk about arrogance.

This example that has been making rounds on the internet beautifully illustrates the problems with the eclectic pedagogy dictated by the Common Core. The number line is strongly promoted by the Common Core for ordering numbers – fractions, decimals, integers, mixed – on it. It is actually well suited for that purpose. But while the number line can be also used in those “strategies” to explain the concept of addition and subtraction, it is ill-suited for doing the addition or subtraction itself – it is tedious, error prone, and not better than counting on one’s fingers. Yet here we have a third grade worksheet, clearly inspired by Common Core’s push for the number line, foolishly used not only to test a student’s actual performing of subtraction, it also expects this third grader to figure out where the notional Jack messed up on his finger counting.

Idiotic problems like this are likely to be found in many “Common Core aligned” textbooks and on the Common Core assessment – after all, they only follow the incessant exhortations found in the standards grade after grade to use “concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship … and explain the reasoning used” rather than simply expect a third-grader to fluently add and subtract. That fluency, Common Core declares, can wait until the fourth grade … while his Singaporean and Korean peers have learned it already in the second grade.

Reader Discussion

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on March 27, 2014 at 15:12:02 pm

Zeev, please explain how you define congruence? The way it is currently done in school mathematics is "same size, same shape". Similarity is "same shape but not necessarily the same size". You should know that the graphs of y=x^2 and y=100x^2 are similar, but they do not look like they have the same shape. To do mathematics you have to know what things mean. How would you define congruence and similarity so that you can work with them other than using translations, rotations and reflections for congruence and add dilations for similarity?

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Richard Askey
on March 27, 2014 at 16:33:12 pm

I agree that many of the so-called Common Core-aligned materials are of questionable value, at best. It is difficult to imagine that they were developed by actual, in-the-trenches, elementary school teachers. However, I thought the CC elementary math was based on "Singapore Math." Is Singapore Math different in Singapore?

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Jeanne Ballou
on March 27, 2014 at 17:15:35 pm

Richard, I am intrigued by your questions; unfortunately I am quite mathematically challenged. In my high school Plane Geometry course (circa 1963), I believe congruence and similarity were described as you have explained. I'm not sure how the CC standards quoted in the article differ from that. When I went to the cited piece (the "track record of failure"?), it seemed that what was being described were properties of symmetry and/or perspective relevant to congruence. But alas, I am definitely NOT smarter than a fifth grader.

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Jeanne Ballou
on March 27, 2014 at 19:10:00 pm

Those are good questions, Richard, but let's not expect poor Ze'ev to play apologist for the writers of CC. And I think the problems in these standards as Ze'ev has quoted them go even deeper. Much deeper. Let alone the poorly defined quality of the definitions students are apparently expected to use, they must now use those notions to PROVE standard criteria for congruence of triangles are equivalent to them. This is a fool's errand. Not only is such a proof either impossible or requires a great deal of reading things into words that are not in those definitions initially, but the task demanded -- at any reasonable level of analytic rigor acceptable in mathematics (here I'm speaking of rigor one would expect at Junior High School level, not honours university math), would be extremely arduous indeed -- it would necessitate abstractly characterizing rigid motions and exactly describing all the possibilities for such motion of triangles, including reflections, and keeping track of where three different points go. I would not want to do this for an audience without careful and lengthy preparation with attention to detail.

I fear that such an exercise is destined to make a mockery of the notion of a "proof". We see students at university (perhaps 90%) who enter our classes with absolutely no concept of what an analytic demonstration looks like, and who require much coaching simply to grasp the logical implications of the phrase "if and only if" in the standard. I have zero confidence that the exercise alluded to will support this concept -- if anything it will give false confidence in an incorrect understanding of it: more un-teaching for us to do when they hit university.

Compare the simplicity and elegance of the Euclidean method for the congruence criteria. Not only is the analysis broken into manageable and understandable steps by the systematic ground-up synthetic approach, but one learns the discipline and rigor of correct mathematical presentation of proofs in the process, which are the "hidden content" of Euclidean Geometry, and what I consider ought to be the most valuable take-away learning outcome from an introductory Geometry course at this level. CC Geometry appears to provide none of that, but rather to undermine some of it, pre-emptively placing incorrect notions where we would like to see correct ones.

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Robert Craigen
on March 28, 2014 at 00:41:08 am


My definition for congruence is very simple: If the shapes are superimposed on each other and overlap perfectly, then they are congruent.

Similarity is like congruence, except that one shape can be (uniformly) scaled relative to the other. After correcting for the scaling, similar shapes become congruent.

I am guessing that we do not differ very much with this definition. Where we possibly differ is:
(a) what is the most effective and meaningful way for 15-16 year old students to prove congruence (and similarity), particularly when they possess rather weak algebraic manipulation skills.
(b) how to operationalize the definition of congruence and similarity for polygons. In my book, congruent polygons match corresponding sides and angles, while similar ones match corresponding angles and have corresponding sides in a fixed ratio.

Having written all this, however, I should point out that I can be completely wrong in my definitions, yet it still wouldn't matter. What would matter is if you could:

- Demonstrate that I am wrong arguing that CC dictates pedagogy.
- Demonstrate that the use of the number line in the example I cited is proper, rather than improper as I argued.
- Demonstrate that I lied when I referred to proofs by rigid transformations as experimental (e.g., by showing some sizable place where they are widely and successfully used in K-12) or as having an established track record of failure (e.g., by showing that Kolmogorov was successful in his experiment with this method in Moscow).


PS Not coincidentally, I also agree with Robert Craigen about the likely damage that proofs via rigid transformations would do the notion of "proof" at high school. Furthermore, I worry that students will leave such a geometry class with the impression that "if things look the same, then they probably are the same" -- precisely the opposite from what we want them to end up with.

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Ze'ev Wurman
on March 28, 2014 at 15:54:51 pm


I fear you are only telling half of the story. I routinely teach my students that congruence means two objects are physically exactly the same. This is to help young students learn what the word means generally. If I want them to prove mathematically that two figures are congruent then I must give them a context and some rules by which this can be accomplished. Teachers have been doing this for years. Euclid was quite good at it.

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Malin Williams
on March 28, 2014 at 16:27:29 pm

There is a dilemma involved in using Euclidean geometry as an introduction to axiomatic reasoning. On the one hand the subject has an intuitive pictorial appeal but on the other hand the logical structure of Euclidean geometry is rather complex
and as well known many of the proofs in Euclid are bogus.

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on March 28, 2014 at 17:52:36 pm

Your geometry example is misleading because it leaves out the rest of the geometry standards. Students are still proving geometric theorems and making geometric constructions, in addition to the example that you gave. They are hardly leaving high school with the impression that "if it looks the same, then they are the same."

I gave the number line problem to my seven year old daughter. She said, "Jack needs to move ten more spaces on the number line." The problem is hardly complicated. It does require an understanding of what subtraction means - something that I want my children to have.

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Samantha Johnson
on March 28, 2014 at 21:14:52 pm

Those who find the transformation-based Common Core approach interesting and full of possibilities, a recent paper by Guershon Harel in the Notices, where he gently(?) dismantles this approach may serve as a nice wakeup call

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Ze'ev Wurman
on March 29, 2014 at 13:36:07 pm

I just need some clarification about the number line problem. It seems that what Common Core Standards are attempting to do with such a problem is ensure that children understand the concept of place value. Place value is a core concept that should be taught along with addition and subtraction.

Jack's problem is simple. He uses the number line to count by 100's three times. Then he gets confused and counts by 10's 6 times, when he should have only counted back by 10's one time. The 6 was in the one's place, so after counting by 10's one time, he should have counted by ones 6 times to get to the correct answer. What did you mean your daughter just counted backwards ten spaces? Did she count by 110's 10's or 1's? Jack stopped on 57; even counting backwards by 1's ten times wouldn't have gotten him anywhere near the correct answer. Were you referencing a different number line problem?

Place Value and other concepts are being missed in school, but standardized testing won't solve the problem. Why? Because understanding place value has nothing to do with the "number-line vs. traditional method of subtraction. True education is leading a student to hone into the key concept and be able to apply it in different instances; like place value on a number line and place value while subtracting. Standardized testing doesn't teach conceptually because it ALWAYS focuses on memorizing the application of a concept instead of understanding the actual concept involved. Hence the need for an interpreter for each concept Common Core is testing application on.

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on March 30, 2014 at 02:03:25 am

Dear Zeev,
Take a look at Grade 8 in the Univ. of Chicago translation of Japanese books for grades
7,8,9. They define congruence and similarity as follows: "Moving a figure from one position to another without changing its shape and size is called a Euclidean transformation. There are three distinct types of Euclidean transformations:
(1) Moving a figure so that all the points on it move a fixed distance in a fixed direction.
(2) Rotating a figure about a fixed point as its center of rotation.
(3) Flipping a figure over a fixed straight line as its axis.
These three transformation are called translation, rotation, and reflection, respectively. Any Euclidean transformation of a plane figure can be accomplished by combining these three types of transformations.

If there are two figures on a plane, and we can lay one on top of the other by moving it, these two figures are said to be congruent.
Pictures are given to illustrate the three transformations, and of two triangles which are shown visually to be congruent with first a translation, then a rotation and finally a reflection used in this order.
This series was edited by a great Japanese mathematician, Kodaira, who returned to Japan from the US where he had been a professor at Princeton and Stanford, to help save the Japanese from the worst of the New Math excesses. It was the most conservative series and was also the most popular one in Japan in one ten year adoption cycle.

The Russian books you mentioned started congruence in grade 6, similarity in grade 7, and asked for much more formal work than either the Kodaira series or the Common Core Math Standards. The most complicated proof in the Kodaira 8th grade part in geometry is to fill in a sketched proof of the theorem that for a given triangle with vertices A,B,C, if the midpoints of the sides AB and AC are D and E respectively, prove that DE is parallel to CC, and that the length of DE is 1/2 the length of BC. In the Common Core that is a high school geometry theorem, not an eighth grade one. I claim that the use of the definition of congruence is a mathematical choice in the CCSSM, not a pedagogical one. A pedagogical choice for the theorem mentioned about would be how this theorem was approached. In the Japanese book there is a helpful drawing with some auxiliary lines drawn, and two specific questions students are asked to prove. I think this is a good choice, but there are other possibilities a teacher might use. That is not in the CCSSM.

The subtraction problem you posted is indefensible, and I do not know how to answer it since I do not know what was given to the student. Some of the material is typed, the location of the right hand point and the three points to the left of it which are 100 apart, but from then on the numbers were written. What was the student given? Has anyone tried to ask the father if he had asked the teacher what the problem really was? I have books from Singapore (in English which is the language they teach in) and Japan (translations) and in both cases addition and subtraction is initially not done in a vertical form, to use the phrase in the Japanese second grade book I have. However, it is translated to a written vertical form and there are problem asking students to correct mistakes. Here are two addition problems on page 56, not quite halfway through the book for the first half of the year.
1 27 2 81
+43 +58
--- ----
60 149

The US teacher used very poor taste in giving the problem which you posted. If the Common Core can be faulted for this it would be the lack of a push for much more serious professional development than has happened. However, without the Common Core there would be even less professional development, which is needed with or without the Common Core. To illustrate this need, take a look at the results of an eighth grade TIMSS fraction item asking which of four ways one would use to compute 1/3 - 1/4. Some data for a few countries, states and Canadian provinces is given at


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Richard Askey
on March 30, 2014 at 14:09:01 pm

Just to add to Dick's comments about that viral Facebook problem: the Common Core does not require the number line method displayed in the problem, whereas it does explicitly require fluency with the standard algorithm, the method that that frustrated parent wants. No previous set of state standards required that. So calling this a "Common Core problem" is a reversal of the truth. The parent, who lives in Indiana, had better hope that the new Indiana standards retain the Common Core's emphasis on the standard algorithm. Since in writing the standards we had to resist a lot of pressure not to require it, I think that requirement is now in danger in Indiana.

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William McCallum
on March 30, 2014 at 16:23:19 pm

There are some references people may wish to consult regarding the implementation of the CCSSM:

Pre-Algebra (Draft of textbook for teachers of grades 6-8) (April 21, 2010)

Teaching Geometry According to the Common Core Standards (For teachers of grades 4-12 and educators)

Teaching Geometry in Grade 8 and High School According to the Common Core Standards (For teachers of grades 8-12 and educators)

Teaching Fractions According to the Common Core Standards (For teachers of K-8 and educators) http://math.berkeley.edu/~wu/CCSS-Fractions_1.pdf

In the second reference, there is this passage:

The truth is that the school geometry curriculum in TSM has been dysfunctional
for far too long and the needed restructuring is way overdue. The new course charted by the CCSSM will be seen to be not only mathematically defensible but also a conservative one, in the sense that it does not inject any new topics into the standard curriculum. Its main innovation lies in nothing more than exhibiting new connections among the existing topics to clarify their mathematical relationships. Let it be noted explicitly that

the CCSSM do not pursue transformational geometry per se.

Geometric transformations are merely a means to an end: they are used in a strictly utilitarian way to streamline and shed light on the existing school geometry curriculum.

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Hung-Hsi Wu
on March 30, 2014 at 18:01:16 pm

Since I have drawn the attention of such mathematical luminaries as Dick Askey and Hung-Hsi Wu, and even Bill McCallum, nothing mathematical I will say — being just a dumb engineer — is bound to sway them. Hence I will quote some pieces from that recent Notices paper by Guershon Harel that all of them so studiously ignored (emphasis mine).

A close analysis of the narrative of the CCSS in high-school geometry revealed potentially serious problems with their future implementations. Our concerns were further validated by some of the initial curricular material developed, presentations given by teachers and curriculum developers in teacher conferences, and conversations with teachers who have participated in regional professional developments targeting the CCSS-Geometry. Collectively, these materials and activities represent a particular interpretation to the CCSS-Geometry that is pedagogically unsound.

Our main concerns can be summarized as (1) lack of attention to students’ intellectual need, (2) premature introduction to and overemphasis on plane transformations, and (3) lack of clarity about the importance to separate between the analytic study and the synthetic study of Euclidean geometry.

Harel poignantly and pointedly reminds us of Poincare writing over 60 years ago: Over a century ago, a great mathematician, with deep pedagogical sensitivity, pointed to the challenge of definitional reasoning: “What is a good definition? For the philosopher or the scientist, it is a definition which applies to all objects to be defined, and applies only to them; it is that which satisfies the rules of logic. But in education it is not that; it is one that can be understood by the pupils.” Poincare’s warning was ignored by the New Math in the 1950s and is being ignored now.

Harel summarizes: The [Common Core] standard approach, thus, would require enormous effort and time to be spent on plane transformations—their definitions, compositions, and properties—which will inevitably shift the attention from deductive reasoning, the main objective of the CCSS-Geometry. … Compare, for example, the insight one gets from a synthetic proof of a concurrency theorem (e.g., “The three medians in a triangle are concurrent”) to the insight one gets from an analytic proof of the same theorem.

Effectively Harel says that Common Core’s shift to transformations-based geometry is counter-productive in its effects even if successful, and that students just beginning geometry are mathematically unprepared for it anyway. Harel does not venture into the parallel ill-advised shift that CC forces with its push for the so-called Functional Algebra that is bound to weaken CC-educated students’ ability to tackle those analytic transformations even further. That is, essentially, what is also said by Jim Milgram, certainly a no lesser mathematical luminary than those who chose to respond here.

But the arguments regarding the transformations-based depth of understanding required by CC Geometry as offered by both Askey and Wu are telling. Both essentially say they envision a superficial approach (“strictly utilitarian way” in Wu’s mellifluous description), to transformations that will not overtax American students. Translation: we will use arm-waving in high school Geometry because that’s the best our students can be expected to do. As opposed to some above I don’t like quoting myself, yet I will: students will leave such a geometry class with the impression that “if things look the same, then they probably are the same” — precisely the opposite from what we want them to end up with.

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Ze'ev Wurman
on March 30, 2014 at 18:04:43 pm

I clearly messed up an html tag at the end of the 3rd paragraph above ... sorry about that. But I am not going to re-post all of it.

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Ze'ev Wurman
on April 01, 2014 at 23:03:47 pm

Dear Dick,

In my previous reply I did not directly respond to your citations from Kodaira because I did not have the book handy and my recollection of it was hazy.

Kodaira’s approach is, indeed, as you described it, with a minor omission that you forgot to mention. Kodaira uses that text you cited to introduce the notion of congruence on a single page, and that is all his grades 7-9 program has ever to say about transformations. Kodaira then happily goes on and establishes triangle congruence based on what I would call intuitive SSS classical construction. Transformations seem not be mentioned ever again, not even for similarity.

So if you want to offer empirical support for successful use of transformations in teaching beginning geometry, I fear you will need to look elsewhere.

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Ze'ev Wurman
on April 25, 2014 at 15:24:56 pm

The infamous Common Core number line was also required for 6th grade Math to find mean, mode, and median, and percent! A useless waste of time.

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