Common Core and Programs Used in “High-achieving Countries”

In debates about the Common Core State Standards for Mathematics (CCSS), one sometimes sees the following type of claim:

This is fully two to three years behind what is expected from students in high-achieving countries.

For example, this claim is made for Common Core standards for addition and subtraction with whole numbers, multiplication with whole numbers, and division with whole numbers (see pp. 16, 17 here).

But a look at “standards” (e.g., courses of study) and textbooks of countries commonly considered high-achieving does not show these gaps.

A Vampire Statistic About Math Anxiety

“93 percent of Americans report experiencing math anxiety.”

Various forms of this statement are circulating around the internet. For example, a search on “93 percent math anxiety” reveals:

Many people (93% according to one study) suffer from math anxiety that stems from the rigid structure of math curriculum and fear of being wrong.

93% of people in the U.S. have math anxiety.

Like a vampire, this “statistic” has no age.

California High School Course-Taking

In California high schools,

• There is a slight male–female disparity in mathematics course-taking.

• There is a large male–female disparity in computer-science course-taking—in the opposite direction.

Here are some statistics . . .

Theory of Arithmetic

Liping Ma and I have written two articles that describe a unified view of arithmetic in elementary grades.

The theory has two sections. The first concerns whole numbers, and the second fractions. Important features of the first section are the definitions of number, unit, like number, sum, and product. In learning arithmetic with whole numbers, students’ initial conceptions of each are established. In learning multiplication with whole numbers, students’ conceptions of unit expand to include many-as-one unit. These features are described in our open-access article here.

The second section of the theory concerns fractions. Students’ conceptions of unit expand again to include fractional unit. The underlying definition of number—a collection of like units—remains the same, but students’ conceptions of number expand to include fractions and their conceptions of unit expand to include many-as-one fractional unit. The underlying definition of sum (and addition and subtraction) remains the same. In multiplication with a fractional multiplier, students use the conception of many-as-one fractional unit. These features are described in our draft article here.

Update: The article on fractions has been published in the open-access Asian Journal for Mathematics Education.

Easter Eggcorns from The Math Myth

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.

The Tyranny of the Mean

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.”

Grade 4 and 8 NAEP Objectives Outside the Common Core

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).

Save the Phenomena!

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.

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

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.”

Unfinished Business from 300 BC

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.)