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…
We admittedly have less tolerance for peculiarities in questions posed to students in Grade 3, the first year of required testing under NCLB, who may not as yet be battle scarred from standardized testing.
To begin, the setup confusingly tallies each class’ “painted tiles” as “cards”. To us, tiles are brittle ceramic squares, cards are thick paper in a rectangular shape. Disadvantage: students.
Part A is a gratuitous Grade 2 level question, so we’ll skip to Part B. We don’t like the dual use of the letters A, B and C to delineate both classes and parts of the question. It’s possible a student might create a “rectangular array” from Class B’s 14 cards/tiles for Part B instead of the 48 total cards/tiles, especially because the question doesn’t clearly link the information from Part A to Part B.
The question also assumes students will find the only pair of factors of 48 that fit into the diagram, specifically, 8×6 or 6×8; however, Common Core standard 3.OA.x isn’t about factoring, it’s about multiplication and division: operations that involve finding one number when two are given, e.g., “interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each” (multiplication) and “interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares.” (division).
Factoring, where you’re given one number and have to find two others (a significant distinction), i.e., where you find various pairs of numbers that multiply to a given number, is not a Grade 3 concept. It’s explicitly written in 4.OA.4: “Find all factor pairs for a whole number in the range 1–100.”
Thus Part B is not Common Core aligned.
Finally, there’s Part C, which is not tied to the numbers either in Part A or B or the initial setup, but is a completely new scenario in which students may backtrack to use the diagram in Part B, since that’s the only place a diagram appears. Part C is aligned to 3.OA.8, “Represent these problems using equations with a letter standing for the unknown quantity.”
This returns us to Common Core and its frequent shortcomings in grade-appropriateness. Grade 3 is not the place for letters representing unknown quantities. That’s a fairly advanced and abstract concept. Third graders should be using small square boxes for unknowns.
In Grade 3, there is Common Core standard 3.NF.2, “Understand a fraction as a number on the number line”, which is one method to ascertain a fraction’s value and also for comparing the values of two or more fractions. (We don’t think many students will have difficulty comparing ⅓ and ⅔, for instance.)
In Grade 4, where fractions with different denominators are first compared (4.NF.2), and where mixed numbers are introduced (4.NF.3c), there is no comparable number line standard to 3.NF.2. We are perplexed by Common Core’s omission of an explicit standard to facilitate students’ expanded understanding of an important visual means of comparing the value of fractions. There is no standard, which means there will be inadequate practice, for students to directly compare (and reconcile the similar appearance of) fractions such as 3/4 and 4/3.
Such a lack of fluency is one reason Grade 4 students on the 2009 NAEP were stymied by this question:
Only 25% (= guessing) of the students got this right, but the most students (41%) chose C, which ironically is the furthest in value from ½, and more (26%) even chose D.
There is an endemic problem in students’ inability to value fractions which Common Core has ignored.
Now to the PARCC task:
This question makes a game attempt at guiding students to make a fraction comparison, but is suited better to a classroom exercise than as an assessment. In other words, students should practice various means to compare fractions and determine fractions’ values (including number lines), but a test should not dictate the method that students use. Some students will prefer to find a common denominator, and the most conversant students will collectively roll their eyes at this simplistic question, recognizing immediately that 3/2 is greater than 1 and 5/6 is less than 1, and won’t have to jump through the hoops this question requires.
We also don’t understand the use of two separate students and two separate number lines in Part A, when a single number line would suffice. The two snippets of number lines are not the same portion, are not horizontally aligned, and rather than facilitate the comparison, hinder it.
Part C does not allow for the entry of the number “1”, which for any discerning student, will be the most obvious choice of answer.
Volume is additive. That, in a nutshell, is the Common Core standard that this question seeks to assess. Now what about the question? In the first diagram, it’s not obvious that there’s a single tank. It looks like two separate tanks next to each other. Why is there a solid line separating the regions, instead of a dotted or dashed line? And why is Part A another Grade 2 addition question? And why is the diagram “not to scale”, a diversionary tactic often employed on the SAT? Do Grade 5 students even know what “scale” means yet?
In Part B, which is “the first tank” to which the question refers? Is it the tank in Part A, which would make the “?”=96+36–18=114? Or is it 18, since the bottom rectangular prism in Part B would have the same volume as the top prism? Either way, it’s ambiguous, and either way, the underlying mathematics in both parts is Grade 2 arithmetic.
Again, poor wording obfuscates the underlying mathematics, which is sound. In Part A, there are missing steps. It’s not obvious that the ratios of Brass to Percussion in Bands 1, 2 and 3 are all the same. That should be posed as an intermediate question; Mr. Ruiz’s realization does not refer either to a particular band or to the total, and it shouldn’t be left up to the student to infer this or to have the same insight as the fictitious teacher. The sentence should be explicit about what it wants.
Part B involves a significant number of steps in the calculation, appropriate for Grade 6, but the task is also not explained clearly. It does not explain that all “210 students who are interested in joining the marching band” will actually be in the band, so that 210 is the starting figure to be allocated among band sections. We gather that the sought after calculation is 210 × 0.8 = 168; then 168 is apportioned in the ratio 3:1, or 126 Brass (and 42 Percussion). However, it won’t be clear to the test taker that 20% of the students, or a different group of 42 (=0.2 × 210, or 210 – 168), won’t be either brass instruments or percussion. The problem does not explain how a marching band is comprised.
Word questions: the Achilles’ Heel of standardized exams. When the descriptive “one-inch frame”, a common algebra scenario, is used in a word problem, it must be indicated in the diagram what that measurement describes. In other words, there should be markings that show the frame is 1” thick on all four sides.
Once upon a time, it was contingent upon the student to take such a diagram and put together the pieces: the frame’s length is (x+2)+1+1; the width is x+1+1. Then students would be asked to solve something, like the area of the picture, or the area of the frame (for which the expression in this problem is simply given (!)). With these computer-based assessments, this problem, as are others, is being spoon-fed to students like bites of Jello.
If the picture’s length is shown in the diagram as (x+2), why do students need to drag the expression (x+2) into the box under the descriptive “the length of the picture alone, in inches”? It sounds like a pre-school activity where children put square pegs into square holes.
One thing we know, this is not high school caliber mathematics: students are not being asked to solve anything.
Function notation is difficult to grasp, especially when its usage varies. For instance, in middle school, students learn y=2x denotes the linear function with slope 2 that intersects the origin, but in later years it’s sometimes written f(x)=2x. The notation becomes truly perplexing because sometimes we write y=f(x) to indicate the general form of an explicit function, i.e., that we have y on the left and x, but no other variables, tied up in various expressions on the right side. Mathematics majors better understand these various forms, but high school students? Not so important. What high school students should understand is functions’ behavior.
Parts A and B of this PARCC task are a high school mathematics equivalent of a London fog, not because of the mathematics, but because of the notation. In Part A, what exactly is a “solution”? Is it an x-coordinate? Or is it the coordinates of a point? Well, Part B indicates it’s looking for f(x), which we now know is the same as y, so that means Part A must have just been looking for the x-coordinate. To further confuse, why does Part B ask only for the value(s) of f(x), when the value of g(x) must be the same at those point(s)?
Lost yet? We’d understand; this is not easy to explain in writing.
What Parts A and B are both driving at is that these two functions intersect twice, as shown in the graph, and therefore there are two solutions to f(x)=g(x), and what are the approximate coordinates of the points at which they intersect? The simple mathematics is obscured by the notation.
The problem never delves into the true high school caliber mathematics; that is, if a student actually forms the equation 1–x=(0.11/x²) and solves it, there’s a quadratic involved [correction: cubic, see comments], which explains the two solutions for x.
For a more articulate discussion than we’ve provided of the confusing nature of what it means to find solutions of a system of functions, read this blog post by Bowen Kerins about an SBAC sample question.
We cannot easily keep in check the Pearsons of the world, with their fingers in every pie, whose sole interest is in profiting from the education market, and thus flood the market with any junk they can get away with, but as for PARCC, SBAC, and the individual States, these are what we must remain most vigilant to oversee, for without a watchful eye, they are also liable to run amok.