Saturday, March 30, 2013

Common Core Conundrum: Absolute Value

I should begin by first disclaiming that while I'm generally pro-standards, I'm also somewhat agnostic about standards. How can that be? Without going into much detail, I believe (a) what we "count" as mathematics is socially determined and standards documents are just part of that determination, (b) I will never perfectly agree with a standards document, but the amount of agreement should not be underestimated, and (c) if there's any power to standards, it's in how they're implemented -- and "good implementation" is very likely to be seen as just "good teaching" under a different set of standards, or no standards at all. It is with this mindset that I watch with some amusement (and sometimes, disappointment) arguments against the Common Core State Standards because of some poorly designed accountability measure. I'm pretty sure poorly designed accountability measures would be a concern right now regardless of the standards in place.

Still, I occasionally see things in the Common Core math standards (CCSSM) that make me stop and wonder, "How are teachers going to deal with this?" One recent instance of that was with how the CCSSM addresses the topic of absolute value. As I looked through Discovering Algebra: An Investigative Approach, I saw the typical "distance from zero" definition followed by an investigation that included this rather unhelpful picture and caption:

(Clearly portrays Elvis? The middle guy looks like Matthew Perry joined a Vegas lion-taming act.)

From there, Discovering Algebra presents students with equations to solve, such as:

$$ | x - 2| + 7 = 12 $$

But do the CCSSM call for solving these kinds of equations? I went searching through the CCSSM for all mentions I could find of absolute value, and here's what I found.

Grade 6


CCSS.Math.Content.6.NS.C.7 Understand ordering and absolute value of rational numbers.
CCSS.Math.Content.6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
CCSS.Math.Content.6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that –3oC is warmer than –7 oC.
CCSS.Math.Content.6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
CCSS.Math.Content.6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
CCSS.Math.Content.6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

and

CCSS.Math.Content.6.SP.B.5c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

If all goes according to plan, there shouldn't be a need to define absolute value in 8th or 9th grade algebra, as it will already have been introduced in 6th grade. I know there are concerns about content being "developmentally inappropriate" for lower grade levels in the CCSSM, but without any personal data to the contrary, I imagine students could come to understand absolute value along with understanding positive and negative numbers. The last standard above, from the moment I first saw it, has been a point of fascination for me. How will 6th grade teachers help students learn topics like interquartile range and mean absolute deviation? You need absolute value for mean absolute deviation, but I don't think that will be the hangup for those who struggle with that standard.

So where else is absolute value mentioned?

Grade 7


CCSS.Math.Content.7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

and

CCSS.Math.Content.7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

Now we have absolute value as it relates to subtraction, and we get another mention of mean absolute deviation in the 7th grade statistics and probability standards. There's nothing yet about solving equations with absolute value or graphing absolute value functions.

Grade 8


Nothing. No mention of absolute value or mean absolute deviation.

High School


The CCSSM doesn't specify what specifically belongs in 9th grade versus other grades, but here's what I found across the whole of the HS CCSSM standards.

CCSS.Math.Content.HSN-VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

CCSS.Math.Content.HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

CCSS.Math.Content.HSF-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

The first of the three standards above, I think we can agree, goes well beyond what we'd expect to find in a traditional Algebra 1 class. The other two standards address the graphing of various kinds of functions, and depending on the activity might belong in either a traditional Algebra 1 or Algebra 2 course, or more likely, both. But do you notice what's not there? Solving equations with absolute value functions. Sorry, \( | x - 2 | + 7 = 12 \), but it looks like you didn't get invited to the CCSSM party. Even if you interpret the second standard to include solving absolute value equations, it only asks for an approximation or to use technology.

Conundrum

The researcher in me wonders what Algebra 1 teachers will do when they get to a lesson on absolute value. Will some just teach it out of habit? Will some think it was excluded due to a CCSSM oversight? Will some modify the lesson to emphasize the graphing of absolute value, but not solving absolute value equations? How many will not even notice it went missing from the standards? Will anyone be willing to kill the little darling?

For years we've lamented math curricula in the United States that was "a mile wide and an inch deep." Given a fixed amount of time, going deeper means going narrower, and there is evidence that the CCSSM supports that (Porter, McMaken, Hwang, & Yang, 2011). For reasons I perceive to be mostly political, the Common Core State Standards don't go out of their way to specifically address content omitted compared to previous standards documents. In 1989, NCTM made a list of areas of emphasis and de-emphasis and many an argument in the math wars was waged using false dichotomies rooted in those lists. I'm sure those who wrote the CCSSM didn't want a repeat of those arguments, so now math teachers and those who support them are left to dig through the standards and find these omissions on their own.

(By the way, if you're looking for an interesting lesson for understanding and graphing absolute value, NCTM is offering one from a recent Mathematics Teacher for complimentary download: Angela Wade's "Teaching Absolute Value Meaningfully.")

References

Porter, A. C., McMaken, J., Hwang, J., & Yang, R. (2011). Common Core standards: The new U.S. intended curriculum. Educational Researcher, 40(3), 103–116. doi:10.3102/0013189X11405038

Monday, March 4, 2013

RYSK: Gutiérrez's The Sociopolitical Turn in Mathematics Education (2013)

This is the 18th in a series describing "Research You Should Know" (RYSK).
"Regardless of the focus of a research project, the fact that mathematics is a human practice means it is inherently political, rife with issues of domination and power, just like any other human practice." (Gutiérrez, p. 40)
This quote from Rochelle Gutiérrez is not significant because it represents a cutting-edge perspective in mathematics education research. Instead, the quote is significant because the "sociopolitical turn" has taken the field of mathematics education research to a place where the above -- directly addressed or not -- is accepted by most math ed researchers. Insisting that mathematics education is somehow politically and culturally neutral is now the marginalized view. We didn't reach this perspective overnight, and we will struggle with how this sociopolitical perspective affects mathematics education. But in return for that struggle, we give ourselves a perspective from which to better understand why some students and some reforms succeed while others do not.

I'd seen a pre-print version of this article posted on the NCTM website leading up to the publication of this year's Special Equity Issue. NCTM has posted the article as a "free preview," which is where Bryan Meyer (@doingmath) found it and asked if anybody would want to discuss it with him. A few weeks later, Bryan and I met via Google+ Hangout and talked about the article.


It's been a few weeks since Bryan and I talked, but my main recollections are (a) my explanation of the socio+political was a bit long, but okay, (b) Bryan's pretty dialed in to this, as evidenced by the quality of the questions he asked, and (c) my speculation of what all this means for day-to-day classrooms gets pretty shaky. On that last part, I stand by my statement that I don't think this means there's something wrong with the sociopolitical perspective. Instead, I think it means I'm still slowly coming to understand it and consider all of its implications. Bryan and I are interested in having more of these talks with more people, and recently on Twitter we threw out a few links for possible articles to read. I'd like to have these Hangouts with more people, so we'll be sure to plan ahead and even seek out a regular time (once a month?) to discuss some recent piece of mathematics education research or commentary.

Feel free to comment to this post, the Google+ event, or the YouTube video. (Here would be nice.)

References

Gutiérrez, R. (2013). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 44(1), 37–68. Retrieved from http://www.nctm.org/uploadedFiles/Journals_and_Books/JRME/articles/JRME_Special_Equity_Issue/jrme2010-08-5a.pdf