Mathematics and Education

A slow blog

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

The Tyranny of the Mean

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In the appendix footnote 3 of a forthcoming 2015 article, Ceci and Williams state:

Many commentators have opined that female scientists are superior to their male counterparts, and therefore the fact that they are hired at the same rate as men obscures the fact that they should be hired at even higher rates, if merit was the basis for hiring.

Ceci and Williams give three quotes intended as examples of this argument. One comes from my blog post The Pipeline and The Trough. However, as I stated in a later post, I did not give this argument.

So why do Ceci and Williams think I did? The answer may lie in “the tyranny of the mean”—the assumption that the mean for a set, e.g., “female scientists,” is the same as the mean of a subset, e.g., “female applicants for a given job.” Read the rest of this entry »

Written by CK

October 19, 2015 at 4:22 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

A Female Advantage in Academic Hiring?

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In their recent Proceedings of the National Academy of Sciences article about tenure-track hires, Williams and Ceci say:

A number of audits of hiring by universities have been reported in the past two decades and these have reported either a neutral playing field in non-mathematical fields . . . , or, more commonly, a pronounced female hiring advantage in math-intensive fields. . . . Here is what we know about the female advantage in real-world hiring of tenure-track applicants in STEM fields in the United States and Canada: There is a female advantage in all large-scale studies dating back to the 1980s. (SI Appendix, p. 26, emphasis added)

Williams and Ceci quantify “female preference” as the ratio of female hires to female applicants. However, they do not compute these ratios for the “audit studies” they cite. This post makes some of those computations.

Interestingly, three of the eight studies cited come from Canada, but some large-scale audit studies of United States universities are not mentioned. This post examines a few of the studies that could have been cited, finding that the claim above is not supported and offering an alternative explanation for the statistics in the US audit studies.

Update (August 2015): Ceci makes an assumption explicit (see comments section here and further discussion below).

Iqbal 2000 Fig 1Update (July 2015): The three Canadian studies concern hiring in the 1980s and 1990s. During this period, the gender distribution of Canadian faculty hires may have been affected by a “brain drain” to the United States. Studies note that PhDs were overrepresented among these migrants. What is known suggests that a very large proportion of these PhDs were male and among “the best and the brightest.” Thus, faculty hiring patterns in Canada and the United States may differ. Moreover, an exodus of men may be a factor in the apparent overrepresentation of women as hires in Canada. Further discussion is below.

Figure from Iqbal, M. (2000). Brain drain: Empirical evidence of emigration of Canadian professionals to the United States. Canadian Tax Journal, 48(3), 674–688.

Read the rest of this entry »

Written by CK

May 11, 2015 at 3:14 pm

Save the Phenomena!

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Last Halloween, the psychologists Wendy Williams and Stephen Ceci wrote an op-ed in New York Times claiming that “academic science isn’t sexist.” Among other things, they suggest that bias doesn’t occur in hiring, writing of “alleged” hiring bias. In a longer article, Ceci and three co-authors claim that “the evidence in support of biased hiring as a cause of under-representation is not well supported, and even points in the opposite direction.” The same evidence is interpreted as a “female hiring advantage” in Williams and Ceci’s “2:1 Faculty Preference for Women on STEM Tenure Track.”

I think the reason why this evidence “points in the opposite direction” is that Ceci et al. do not “save the phenomena” by accounting for crucial details of the findings they cite. Thus, these findings may not be consistent with the findings of Williams and Ceci’s experimental study. This raises concerns about the ecologically validity of the experimental study, e.g., that it may be not realistic to assume that a strong female applicant will often be described as “creative” or “a powerhouse.”

Here are details. Read the rest of this entry »

Written by CK

April 14, 2015 at 4:33 pm

Who is the Most “Mathematically-intensive” of Them All?

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Well, mathematics, of course. But what comes after that? Engineering, computer science, or economics perhaps? Answers differ, even according to the same definition of “mathematically intensive.” Read the rest of this entry »

Written by CK

November 7, 2014 at 2:10 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