Mathematics and Education

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Archive for the ‘Common Core State Standards’ Category

Easter Eggcorns from The Math Myth

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In her article for Slate, Evelyn Lamb points out that parts of Andrew Hacker’s book The Math Myth are not exactly Easter eggs—although they function as jokes for numerate people, their humor seems unintended.

Instead, they might be dubbed “Easter eggcorns.” The word “eggcorn” arose from “acorn” and now means:

A series of words that result from the misunderstanding of a word or phrase as some other word or phrase having a plausible explanation, as free reign for free rein, or to the manor born for to the manner born (from William Shakespeare’s Hamlet). American Heritage Dictionary

Thus an “Easter eggcorn” combines the hidden surprise of an Easter egg with the misunderstanding and plausible explanation of an eggcorn.

Along with Easter eggcorns for numerate people, The Math Myth has joke-like surprises for people familiar with K–12 mathematics education in the United States. (For my background in mathematics education, see this page.) This post is devoted to examples of the latter. Because parts of these examples may seem like misquotations or misreadings, I have included scans of pages from the book.

Standards are test questions?

Chapter 1 begins with a brief discussion of concerns about mathematics education in the United States. It says that “calls are heard to return to rigor and end feel-good rostrums” (p. 5). We are told:

Scan p.5(bottom of p. 5)

Scan p.6(top of p. 6)

As noted several chapters later in The Math Myth (Chapter 8, p. 124), these “questions” are “selections” from the Common Core State Standards for Mathematics (CCSS), a 93-page document which can be downloaded here. In particular, they come from two standards for high school (Functions, Interpreting Functions 8b; and Algebra, Arithmetic with Polynomials and Rational Functions 5).

The CCSS are expectations for what students should know, not tests of whether they meet those expectations. Such tests have been developed by two consortia formed by groups of states, Smarter Balanced Assessment Consortium (SBAC) and Partnership for Assessment of Readiness for College and Careers (PARCC). Examples of the types of test questions that students will actually face are here (SBAC) and here (PARCC, Question 9 involves an exponential function).

It’s instructive to compare The Math Myth version of the second “question” with the original:

a-apr 5

There are several differences: the 5 (which is the number of the standard); the footnote; and the plus sign in parentheses “(+).” The plus sign indicates that the standard is “Additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics” (see p. 57 of the Common Core Standards for Mathematics).

Unlike the “plus standards”:

All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students.

Thus the title of Chapter 8 is another Easter eggcorn:

scan chap8

A more accurate title would be:

chap title

Standards are lesson plans?

Chapter 8 begins by describing the Common Core State Standards (CCSS) as being developed under—confusingly—“the aegis of a transcontinental conglomerate called the Common Core State Standards” (p. 117). It refers to the Common Core as “this curriculum” (p. 118, line 1) and states that:

Ultimately, the Common Core morphed into a full set of K–12 lesson plans, spelling out what every teacher was expected to impart and every student to learn. (pp. 120–121)

When I first read the sentence above, I thought “a full set of K–12 lesson plans” was a reference to the outcome of a project (e.g., EngageNY) which provides instructional materials aligned to the CCSS. (An example of an EngageNY plan for a lesson on exponential functions is here.) A typical lesson plan is at least one page long (often much longer), but the document that delineates the Common Core Standards is only 93 pages. Surely The Math Myth did not mean to imply that this document was “a full set of K–12 lesson plans.”

However, I changed my mind several pages later (p. 123). In the same paragraph, the Common Core (I think this means the “transcontinental conglomerate”) is said make assertions about “its lesson plans,” which are countered—according to The Math Myth—by examination of “its 1,386 standards.”

Does this mean that The Math Myth considers a standard to be a lesson plan? It’s not uncommon for standards documents to be misinterpreted in various ways, e.g., as dictating instructional sequence, or being instructional materials.

Making sense (and nonsense) of these examples

Considering the Common Core State Standards as a collection of test questions, collection of lesson plans, or transcontinental conglomerate may have a certain plausibility if each meaning is taken individually.

Although standards themselves are not test questions, they determine the content of test questions. Similarly, standards are not lesson plans, but they set goals to be reached via lesson plans. Although a set of standards is not a transcontinental conglomerate, a transcontinental conglomerate (Pearson) administers the standards-aligned tests created by PARCC.

Taken together, however, these different meanings for “standards” are likely to be unhelpful for readers who are not already familiar with mathematics education in the United States.




Written by CK

July 31, 2016 at 3:43 pm

Grade 4 and 8 NAEP Objectives Outside the Common Core

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This is a guest post by Jason Zimba.

Created in the late 1960s, the National Assessment of Educational Progress (NAEP) today measures U.S. achievement in mathematics, reading, science, U.S. history, and other subjects. The most recent framework for the mathematics assessment is described in a document published in 2014 by the National Assessment Governing Board and entitled Mathematics Framework for the 2015 National Assessment of Educational Progress. Below, I list assessment targets from the NAEP Mathematics Framework that are outside the expectations in the Common Core State Standards for Mathematics (CCSS-M). Read the rest of this entry »

Written by CK

October 15, 2015 at 4:45 pm

Unfinished Business from 300 BC

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The double meaning of “ratio” in U.S. school mathematics is a phenomenon that goes back as far as Euclid. On the one hand, a ratio of two numbers is written as a pair, e.g., 3 to 1. On the other, a trigonometric ratio is a single number, e.g., the sine of pi/6 is one half—not 1 to 2. (Ratios of more than two numbers do not have this double interpretation, so are not part of this discussion.) Read the rest of this entry »

Written by CK

September 29, 2014 at 3:24 pm

Comments on Ravitch’s MLA Speech

with 2 comments

The historian Diane Ravitch gave a speech to the Modern Language Association on January 11 about the past, present and future of the Common Core State Standards which was posted on a Washington Post blog. There’s a lot to like about the speech when it comes to rethinking uses of tests and test scores. I’ve been in favor of caution about testing since at least 1999 (see my article here).

However, the speech has some statements that are unclear, appear unaware of research in mathematics education, or seem uninformed. Some concern:

Characteristics of standardized tests.

Field testing standards.

Developmental appropriateness of the CCSS.

Development of the CCSS.

Details are below. Read the rest of this entry »

Written by CK

February 13, 2014 at 9:09 am

Comments on Goodman’s “Comparison of Proposed US Common Core Math to Standards of Selected Asian Countries”

with one comment

Summary. In July of 2010, Jonathan Goodman published a comparison of Common Core State Standards with curriculum documents from several Asian countries (China, Hong Kong, Japan, Singapore, Taiwan). In my opinion, his analysis has some serious flaws. In this post, I give some examples. In an earlier post, I have given a brief overview of differences in national context, noting the different uses of standards and other documents in the U.S. and elsewhere. These different contexts and uses suggest how a U.S. reader’s expectations may lead to misinterpretation of documents from outside the U.S. In this post, I compare some of Goodman’s statements with the content of these documents in two ways: comments and detailed side-by-side comparisons. Read the rest of this entry »

Written by CK

August 4, 2013 at 4:12 pm

Comments on Milgram’s “Review of Final Draft Core Standards”

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Summary. In 2010 (apparently in June), Jim Milgram posted a review of the Common Core State Standards, comparing them with standards of high-achieving countries. In my opinion, his review misses some important details and makes some incorrect conclusions. In this post, I give some examples. In an earlier post, I have given a brief overview of differences in national context, noting the different uses of standards and other documents in the U.S. and elsewhere. These different contexts and uses suggest how a U.S. reader’s expectations may lead to misinterpretation of documents from outside the U.S. In this post, I discuss some assertions in the review and give some detailed side-by-side comparisons of comments with standards, teacher’s guides, and other documents. Read the rest of this entry »

Written by CK

August 4, 2013 at 3:55 pm

Interpreting Standards from Other Nations: Considerations of National Context

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Over the past decade, comparisons of U.S. standards for mathematics have been made with “standards” from other countries, e.g., national curriculum standards, syllabuses, or courses of study. Some of these comparisons overlook important details, resulting in conclusions whose accuracy could be improved considerably without much additional effort. This post gives a brief overview of two differences in national context that affect interpretation of documents from other countries, in particular, China, Hong Kong, Japan, Korea, Singapore, and Taiwan. (Further details are in an appendix at the end of this post.) The two posts that follow (here and here) discuss comparisons that have been made by (respectively) the mathematicians James Milgram and Jonathan Goodman. Read the rest of this entry »

Written by CK

August 3, 2013 at 1:42 pm