While Smarter Balanced Assessment Consortium has already administered a pilot exam with more than one million participating students, the other state consortium, Partnership for Assessment of Readiness for College and Careers, seems well behind in what is becoming a consortia war, and has already had at least one state jump ship.
To further add to our sense that PARCC is in disarray, information releases coming out of PARCC are few and far between and have this year decreased in frequency. Nevertheless, with little fanfare, PARCC on August 19 released for public review a new series of item prototypes. Two companies, ETS and Pearson, have contracted with PARCC “to develop the items and tasks for the PARCC mid-year, performance-based and end-of-year assessments”, but PARCC does not make clear the source of its latest batch of questions.
With the extra time PARCC and its contractors have taken, we would expect a fairly sizable sample and a certain degree of polish, however, we can already express our disappointment that the total number of newly-added tasks equals only fourteen: one task each in Grades 3, 6 and 8, two each in Grades 4, 5 and 7, and five for high school. While SBAC won’t be assessing students in Grades 9 and 10, PARCC has left this decision open but nonetheless designates high school questions Algebra 1, Geometry, and Algebra 2.
As for polish, let’s have a look.
While we at ccssimath.blogspot.com think it’s never necessary for students to consciously think of “division as an unknown-factor problem” (see standard 3.OA.6), if this standard is going to be assessed, a straightforward mathematical question should not be obfuscated by poor language. We are sorry to readers for our repetitiveness, but poorly worded tasks are going to trip up students who actually can do the math. In particular, if students can calculate 72 ÷ 9, why should they get this question wrong because they don’t quite get the meaning of “the given equation can be rewritten using a different operation”?
This is one of those ridiculous examples of how computer-based mathematics tests can be a giant step backwards. More than the irony of looseleaf paper appearing on a computer screen is a criticism we’ve leveled before: students have to first write this question onto a sheet of real paper in order to solve it.
This is another example of how Common Core is shoehorning students into solving problems in a particular way, which runs contrary to sound mathematical practices. Many students simply do not have to perform this entire rigmarole in order to convert fractions to decimals and to compare decimals. Even if students need to “justify” why 0.2 is greater than 0.17, proficient students likely wouldn’t use a grid to do it. There are better methods.
While we could take exception in this task to odd usages such as “⅚ foot” (who ever uses such a measurement?) and its requiring students to show on the square that they understand what a fraction means (a Grade 3 topic), the main objection we have is to the Common Core standard, 5.NF.4b, which thinks it makes sense to revisit a concept which has already been fully developed and reached closure two years earlier: the area of a rectangle.
It does not further students’ conceptual framework to revisit a topic that was finished in Grade 3 by simply adding the complication that side lengths can now be fractions. There are plenty of other useful applications for multiplication of fractions, so why waste time reviewing such a basic topic? Are students going to rehash this Grade 3 geometry topic every time they learn about a new type of number, such as decimals and irrational square roots?
The end result of Common Core’s spiraling is that a major portion of this task is assessing students on two Grade 3 concepts, the area of a rectangle, and knowing what a fraction represents.
There are many things to find fault with in this task, but we’ll select the most egregious: the use of the number 1 on a number line to represent “the whole”, which in this example is 12 pencils. No mathematically competent teacher would have students solve this problem in class by using a number line demarcated with a 0 and 1; they would use a rectangular bar.
This is a fairly challenging word problem involving common denominators; give students a fair chance to solve it.
As we previously pointed out in our analysis of NYSED’s exams, the phrase “can be used” gives the task author excessive leeway to write a bad question.
Rate problems can very, very difficult, and a sensible teacher will wisely suggest to students to always start with the same formula, R x T = D. It’s far easier to remember and creates a starting point with fewer pitfalls than any other form of this equation. Which is why this problem would be more fairly presented as what is written in Part B. (But then it really wouldn’t take advantage of such cool computer capabilities, would it?)
How would skillfully solving this problem play out?
If a student writes: R x T = D
…the correct equation for this task set-up…
Then plugs in known quantities: R x 12.5 = 100
…the number 100 is not going to move to the left side of the question, no matter what the student does to solve it, even if the student uses reciprocal multiplication instead of division. Although the equation in Part A “can be used” to find R, no one following fairly standard procedure would ever have the equation in that form. This is another example of how Common Core forces students to conform to a particular method, instead of giving them free rein to solve a problem as they see fit.
Although the language is again poor (we Googled “least rate” and got a paltry ~60,000 hits, while the better term “lowest rate” got us ~3.5 million hits), the mathematics underlying this problem is straightforward, sound and challenging.
Students first have to use the formula (reading rate) x (time) = (number of pages read) to calculate each rate and then use those rates to calculate a different time, the time to completion. The second series of calculations is complicated by the extra step of finding the remaining number of pages to read. That’s a lot of calculations.
Although Common Core standard 7.RP.3 states Grade 7 students should be doing multistep problems, it does not suggest that the set-up should be explaining those steps. Grade 7 is the age range for students to start being exposed to more serious and lengthy tasks. Instead of spoon feeding students the steps they need to take and asking them to drag the names to boxes in order (which will lead to blind guessing, we are sure), the problem should be set up and students should be left alone to solve it.
Students should solve these kinds of rate problems in class and they should expect to see similar set-ups on exams.
We examine this problem immediately following the book reading and running problems to compare and contrast them. While this task is posed for Algebra 1, which is traditionally taken in Grade 9, it’s far simpler than the Grade 7 book reading problem but uses the same basic formula (rate) x (time) = (amount). The running problem also uses a variation on this same formula. The pool problem’s only complications are simple conversions between minutes and hours, and that the y-intercept of the graph is not 0, a Grade 8 linear equation concept. Why it’s given to Grade 9 students we have no idea. It’s a step down, not fitting with the increased complexity that high school demands.
Both PARCC and the Standards have a part to play, but of one thing we can be sure: Common Core’s drawn out and convoluted presentation of the overlapping and related concepts of rates, ratios and proportions in Grades 6 and 7, and linear equations in Grade 8 (rehashed as linear “functions” in Grade 9) is not going to lead students to synthesize all of them into an integrated and cohesive whole...an essential understanding that students will need before they start looking at curved functions...and differential calculus.