PARCC re ducks

Many textbooks and other materials are lightly edited and rebranded by their creators as Common Core aligned, but being there is no central ministry of education, as in Singapore, which reviews materials and issues an official government seal of approval, anyone can make such claims with impunity.  Some education departments are making their own determinations, such as NYC, which chose “Houghton Mifflin Harcourt's ‘Go Math’ program for elementary students, and Pearson's ‘Connected Math Program 3’ for the middle grades”, or Louisiana, which last year rejected “every math and reading textbook submitted by publishers”.

The precise wording of the 93-page Common Core State Standards for Mathematics notwithstanding, a lack of consistency in interim assessments, independently developed and posed to students in states such as Kentucky, New York, Illinois and North Carolina, raises the issue of whether these test questions accurately reflect the Standards and manifest Common Core’s intent, but no matter: states, too, are barreling ahead with no independent oversight.

Carol Burris, a principal at a high school on New York’s Long Island, whose essays are often published in the Washington Post blog The Answer Sheet, recently critiqued a math test for first graders and critiqued several sample math questions.  Lest we ourselves become completely overwhelmed by myriad Common Core offerings that run the gamut, we declined to pass specific comment on those independently written questions, and instead continue to focus on states’ sample and/or actual assessments and, to date, sample-only questions designed by the two  “official” consortia, SBAC and PARCC.

This preamble brings us to PARCC’s latest batch of sample items, twelve in total, released in early November, for Grades 3-6 (nothing new for Grades 7 or 8) and high school.  Fasten your seat belt…

Fractions are numbers, too – Part 3

The National Assessment of Educational Progress (``The Nation’s Report Card’’) in 2007 posed the following question:

Sidestepping for now the issue that only half of eighth graders correctly solved an elementary school problem, calculator allowed, the intrigue lies in the distribution of wrong answers:

The Life of Pi

Beware the ides of March and a Happy Saint Patrick’s Day t'ya, but as winter snows abate and the vernal equinox comes nigh, a young mathematics teacher’s fancy turns to thoughts of...π.

Spreading like the ``Harlem Shake’’ meme, it seems every classroom across the land now observes an in-school holiday called Pi Day.  For the uninitiated, Pi Day is cleverly celebrated on March 14, or 3/14.  A quick trip over to Pinterest regales you with endless examples of the hackneyed puns, cartoons, song lyrics, decimal expansions, and other non-activities that grace school walls to mark this annular (HA!) event.

Fractions are numbers, too – Part 2

We have a lot to say about CCSSI’s treatment of fractions, which starts tentatively with 1.G.3, but we’ll initially hone in on Grade 3, which is where Common Core begins its big push.  We’ll discuss Common Core’s sequence, and compare or contrast it to our own preferences for how fraction concepts should be introduced, and if we differ, provide a (hopefully justified) rationale for our choices.

3.NF.1 states, ``Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.’’

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.

Graphs and data analysis – Part 2

What happens to 12 pizza pies at a school party?  POOF, before you know it, they’ve disappeared.

We begin, as we often do, with questions from the National Assessment of Educational Progress (``The Nation’s Report Card’’).  In 2005, students were asked to study a pie chart (alternately called a circle graph):

Graphs and data analysis – Part 1

In 2007, the National Assessment of Educational Progress (also known as ``The Nation’s Report Card’’) gave the following task to 4th graders:

This question demonstrates that even at the elementary school level, it is possible to usefully integrate several aspects of problem posing: that the problem be lengthy, that it bring together multiple math skills, and that the path to the solution not be readily apparent.  (We have previously coined the phrase ``length, connectivity and dimensionality’’ to describe this triumvirate of features.)

As our regular readers may surmise, we wouldn’t lead off with this question if there weren’t something more to it.  The specific task the NAEP required students to perform barely scratches the surface of the issues and learning possibilities contained in this fact pattern. 

Mathematical tools – Part 3

On the morning of June 20, 1944, two weeks after D-Day, New York high school students sitting for the Plane Geometry Regents exam were asked to do the following construction:The task posed almost 70 years ago was not difficult, but it also wasn’t rote.  It required the test-taker to make connections and to solve a problem: to apply basic construction techniques to notions of similarity and its theorems.  Additionally, it required visualization skills: before actually drawing the lines and arcs necessary to complete the construction, students needed to envision the final picture.

Mathematical tools – Part 1

CCSSI 7.G.2 states, ``Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.’’

Some preliminaries before we parse this standard.

Protractors

We here at ccssimath.blogspot.com LOVE protractors.  The way we see it, a (non-toxic, teething) protractor should be given to every newborn in their bassinet (thanks to Charles Schulz for the inspiration); perhaps the first one can be hanging from a mobile.  A protractor (like a banjo) is a happy thing: it’s a smile, or a big letter D.  A protractor is easy to hold, it’s too big to choke on, and it’s covered with numbers.  It even has mystery and intrigue, for why would the numbers run in opposite directions?  Rulers, in contrast, though useful, are dangerous things; they fit in the mouth and can be brandished as swords.

Don't punt; take skills into the end zone

In 2011, the National Assessment of Educational Progress (otherwise known as ``The Nation’s Report Card’’) asked fourth graders to calculate the following:

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The results were 83%, 64% and 52% correct, respectively.  Why was performance on the last task, arguably the easiest of the three, the worst?  Answer: probably because it was the only one in which calculators were not allowed.

The concept of area – Part 3

This is the third part of a multi-part blog post on the concept of area.

Pop quiz!

What’s the formula for the area of a triangle?

The concept of area – Part 1

This is Part 1 of a multi-part blog post on the concept of area.

In 2005, the National Assessment of Educational Progress, otherwise known as ``The Nation's Report Card'', presented the following question:

 

This question was given to fourth graders, 47% of whom answered correctly.  This result would have been mediocre if tiling had been taught by fourth grade and good if tiling had not been taught (i.e., students were able to figure it out on their own)—but we don’t know which.

Wholesale whole-number murder and redemption

An extended mathematics metaphor:

Flowing under the pre-K through high school curriculum, like the ever-widening Mississippi, is a steady expansion of the number system and its corresponding basic operations.

Important milepost concepts and mathematical problems which are posed, deconstructed and solved throughout those years are like the flatboats and steamboats of Mark Twain's era which floated upon the Mississippi's waters.

One doesn't seek to control the river, because the number system and its operations exist in nature, but we can select what floats on it and where to travel.  Choosing when and how to introduce concepts, what problems to pose, and where they fit is the foremost responsibility of a standards and curriculum developer.

How many is a billion?

In 1992, the National Assessment of Educational Progress, otherwise known as ``The Nation's Report Card'', presented the following question:

In this five option multiple-choice question, 22% of American eighth graders got the correct answer.  We could analyze in purely statistical terms how much better than wild guessing this is, but it's clearly not much.  In other words, the results essentially meant that the entire nation's soon-to-enter high school students did not understand the difference between a million and a billion.