Fractions are numbers, too – Part 1

We hold these truths to be self-evident, that all numbers are created equal...
(Well, Abraham Lincoln or Thomas Jefferson could have written this.) 

On February 8&9, 2013, while much of the northeastern US was getting socked with a blizzard, a symposium was held at Educational Testing Service headquarters in Princeton.  The meeting between ETS and the National Urban League was entitled "Taking Action: Navigating the Common Core State Standards and Assessments," and the purpose was to ``discuss [the] impact of Common Core State Standards on underserved communities’’ and ``consider strategies to succeed with the new standards and assessments.’’

We stumbled across the live-twitter feed by accident, but immediately recognized the meeting's significance, as David Coleman, Joe Willhoft, and Doug Sovde, three Common Core ``biggies’’ were all featured speakers.  For them, it offered an opportunity to ``sell’’ CCSSI to important community groups: in addition to the NUL, representatives of the NAACP, NCLR and SEARAC were also in attendance.

Doug Sovde works for Achieve, Inc., ``as Director, PARCC Instructional Supports & Educator Engagement’’.  According to the PARCC website, ``Achieve was selected as the Project Management Partner for the PARCC consortium to facilitate the activities of the Partnership and help ensure the PARCC vision is fully realized.’’

The tweets from the NUL representative mostly confirmed that the speeches by Coleman, Willhoft and Sovde were the usual pitch, but one tweet quoting Sovde went as follows:

We certainly agree with the obvious sentiment that math builds on itself and students who fall behind will have increasing difficulty in later years, but it’s the second sentence that we want to consider.

The assertion brings us back to our previous blog post about how Common Core holds itself out.  CCSSI’s introduction states it is standards, not a curriculum, and does not dictate teaching methods.  A standard would say ``Students will understand fractions and be able to...’’ and if that is what CCSSI is, how can Sovde already give it credit (gazing into the future, no less) for achieving that standard?

We have become accustomed to such propagandizing doubletalk, but we won’t stand for it.

Curriculum writers and teachers shouldn’t let slide such an unsubstantiated credit-grab and parents and community groups shouldn’t, either.

***

Nevertheless, it is true that fractions and algebra present two of the most timeless fundamental skills hurdles for students, and so, we here at ccssimath.blogspot.com are particularly concerned with how (and, indeed, whether) CCSSI makes a game attempt to achieve real reform.  According to Sovde, Common Core has already succeeded, at least with fractions, but we are not such good prognosticators.

As is our wont, we will first try to diagnose the difficulties underlying fractions, then see what CCSSI does about it, and if it’s not enough, present some of our own ideas.

In 2003, the National Assessment of Educational Progress (``The Nation’s Report Card’’) asked the following question:

That 94% of eighth graders could recognize the correct picture is no small relief, and 89% (Grade 8, 2007) can even reduce simple fractions:


Even when posed as items in a group, rather than as a picture, 80% of fourth graders (2007) knew what the numerator and denominator of a fraction represent:

So we are not impressed with how SBAC envisions an advancement in assessment sophistication.  By replacing a multiple choice question...
...with a series of true/false questions...

...SBAC believes that ``a clearer sense of how deeply a student understands the concept of ⅖ can be ascertained.’’

We recognize this sample SBAC question is only an illustration of how it plans to modify traditional assessments (this question is an actual prototype), but it remains to be seen whether such reformulations can more usefully gauge, and therefore lead to improved, student achievement, especially if most already understand the basics.

***

Let us skip forward in complexity to a ``short constructed response’’ problem that involves more than fraction identification and equivalencies. The following NAEP question was given in 2005:

The first two sentences present a scenario no more complex than the previous questions we looked at.  It is the complication presented by 3 people that requires either a calculation, a drawing, or some other intermediate steps, which sets this problem apart.  We have no way of knowing which path(s) to the solution students tried; either fraction arithmetic or using a picture would be a workable method.

We were genuinely shocked at the results: with no restrictions on which method could be employed, only 22% of twelfth graders arrived at the correct answer.

We suspect those very same students would have no problem multiplying ¼ (or 2/8) times ⅓ if the problem asked only for this calculation.  Perhaps dividing ¼ by 3 would result in somewhat fewer correct answers.  If a picture were provided, maybe students would think to draw some dividing lines and get the right answer that way.  But the way the problem was given, requiring more than 1 step, an unacceptable number of American students were stymied somewhere during the problem-solving process.

Somewhere in the progression from understanding the notion of what a fraction represents to working with fractions as numbers in everyday activities and more lengthy analyses, there is a conceptual divide.  We haven’t yet determined where it lies, but we’d venture to say that the writers of Common Core didn’t bother to ask these questions.

As we proceed, we’re going to look for the divide, and address how Common Core, as written, will likely affect the classroom: how teachers teach and how students work with fractions, in particular, how emphases have changed and whether those changes are improvements.

Will the adoption of Common Core change whether students ``get fractions’’?  With 45 states and DC signed on, we can only hope.


Continue to Fractions are numbers, too – Part 2