NYSED's released 2013 exam questions

The New York State Education Department clearly tried to play it safe in designing questions for the Grades 3 to 8 exams, with a lot of calculation and solve-the-equation problems, which would have been the same pre- or post-Common Core.  Where NYSED gets into trouble is when it tries to be creative, particularly in word problems.

So without further ado, here’s a look at some of the released questions from the 2013 New York State Testing Program’s Common Core Mathematics Tests.

Grade 3
 

The weakness here is one of language.  The set-up question uses the word “objects” in order to cover the nouns in all the various scenarios, which include: boxes, pencils, people, buses, marbles, groups, books and shelves.  Within each answer choice, the student first has to consider which “objects” are "described".

We’d be hard-pressed to apply the term “objects” to people, but it’s equally inapt for shelves and buses, so we could eliminate choice B simply because there are no “objects”.  Luckily, it’s not the answer that received credit.

The problem with credited answer C is that while the word "objects" might apply to marbles, but not “groups”, the set-up describes neither.  24 ÷ 4 is the number of marbles in each group.

The question could ask, "Which situation can be represented by 24 ÷ 4?" and the answer would be “The number of marbles in each group when 24 marbles are sorted into 4 equal groups.”  That's unambiguous, grade-appropriate, and aligns with 3.OA.2.  Instead, students are left to infer a question that's not explicitly asked.  The task of making such a language inference is not what the Common Core standard is supposed to require here.  This is a math test.

Although we understand the intent of this problem and the underlying math is sound, the problem is heavily biased against students with language weaknesses because it requires inferences and fairly advanced interpretive skills, especially for Grade 3.

Grade 4
 

Although many students would choose the correct answer by simply looking at the picture and seeing 4 bars and recognizing that each bar is ⅛ shaded, finding the answer by this shortcut avoids the real mathematics: understanding the model and what it represents, which will lead to difficulties for students applying fractions to solve problems in the future.

It’s implicit, but not made clear that each of the four bars is “a whole”.  The weakness reveals itself when one does the actual calculation 4 x ⅛, because the "total shaded area" in reduced form, ½, is not visibly apparent anywhere in the model.

If the answer choices were given as fractions rather than “multiplication sentences”, and two of the four choices were ½ and 4/32, we are certain many students would choose the latter wrong answer (because 4 out of 32 rectangles are shaded), even if the problem began, “If each bar in the model represents a whole...”

Grade 4

The problem here is simply the phrase “wrote a pattern”.  It’s not mathematical and it’s terrible English. Why not teach students good math habits from the beginning and use the correct term “sequence” (which is correctly in 4.OA.5) and phrase the question, “Rosa wrote down a sequence of numbers that contained a pattern.  The pattern followed the rule: subtract 7 from each term in the sequence to get the next number in the sequence.”

Grade 4

The problem here again is the use of language, in particular the multiple and varying uses of the word “number” in one sentence.  The “number below” and “the number represented by the digit in the thousands place” are two very different meanings of the same word.  They are not referring to the same thing.  Disadvantage to weak readers and ELLs.

Grade 5
 

Although NYSED purports to align this question with 5.NBT.1, this question simply does not comport with the standard, which states, “Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.”

NYSED explains, “In this case, the 3 in 3,000,000 represents 10 times what a 3 in the hundred thousands place would represent; equivalently, the 3 in 3,000,000 represents 30 hundred thousands.”

While the first part is certainly true, and 3,000,000 is ten times 300,000, the key gap in NYSED’s thinking is the magical word “equivalently”.  No one says “30 hundred-thousands”, ever.  What’s happening on the right side of the equation is really 30 x 100,000, which is a multiplication problem.  The underlying math is sound, it just doesn’t align, and thus the question tests aptitude, not the standard.

Grade 6

Here is an example of bad mathematics.

While simplifying the expression in choice A leads to the correct distance, 9, one does not calculate a distance by taking absolute values and adding them.  One takes the absolute value of the difference in values.  Thus, the correct expression would be either | 3 – (–6) | or | (–6) – 3 |.

If the points were instead A(–6, 4) and B(–3,4), applying NYSED’s "method", one would get |–6|+|–3|=9, which is wrong.

Grade 7

The weakness in the question is the phrase “can be used” which allows the question writer to get away with (if we didn’t object) to a poorly formed task.  While choice B is the only equation that will result in the correct value of x, which represents the hourly charge, the equation is certainly not the initial equation we would set up to find the answer algebraically.

The term “total charge” implies that the value, $320, should sit alone on one side of the equation, and indeed, that’s the way students should be taught.  Weaker students, who are nonetheless properly conversant with setting up equations, would incorrectly eliminate both choices A and B on that basis.  Choice B is not the properly set up initial equation, it is an intermediate result in the solving of the initially correct equation, which would be either 320 = 5x + 70 or 5x + 70 = 320.

Grade 7

Common Core is not supposed to dictate problem solving methods.  It’s supposed to allow students the freedom to choose their own method and argue on its relative merit or weakness compared to other methods.  That’s one of Common Core’s alleged strengths that its authors trumpet constantly. See SMP3.

Yet in this problem, students are shoehorned into selecting a method that they might ordinarily abhor, or worse, not be able to complete.  And indeed, we the authors of ccssimath.blogspot.com, would never solve this equation by first dividing both sides by ⅔.  It ends up looking like this awful mess...

...where NYSED conveniently glosses over what happens between steps 2 and 3.

This problem would be far better simply if it asked students to solve the equation. Then they can use any method they so choose.

Grade 7

This is one of the most sophisticated questions in the Grades 3-8 exams.  Yet students are likely ill-prepared to take a simple relationship, such as that between the length of a square’s side and its perimeter, and represent that relationship as a linear equation.

This difficulty is compounded in NYSED’s question by its wording, particularly its abundant and overuse of the letters x and y in different ways, which will confuse a lot of students.  But most importantly, what’s missing from the problem is a critical intermediate step: an understanding that y, defined as the perimeter of the square, is related to x, defined as the side of the square, by the equation y=4x.  That equation is at the core of the problem and should be an explicit part, rather than the more ambiguous wording “straight line” which appears.

To those who recognize the complexity of the mathematics of this problem, we offer a question of this sort from the Five Triangles blog, shown below, but the wording and the relationships must be made crystal clear:

Grade 8

Although “grasping the concept of a function and using functions to describe quantitative relationships” appears in Grade 8, functions truly make their formal debut in Common Core’s high school standards, in F-IF’s “Understand the concept of a function”.  This is but one of numerous unreconciled redundancies in Common Core.

This Grade 8 question assesses students’ understanding of how to apply what is more commonly known as “the vertical line test”.  It really doesn’t belong in Grade 8, since a first introduction to functions in any grade should begin with a mapping from one set to another, followed by sets of ordered pairs, followed by linear functions.  Those alone would be enough for Grade 8.  Choice B is particularly unsettling because it’s supposed to represent a quadratic function, which is never studied in Grade 8.  How would students know that the arrows in the parabola don’t become vertical or bend back?  We have no idea.  It’s a bad question for the Grade 8 exam.

Grade 8

This question does not align with standard 8.EE.3, in which students should be able “to

express how many times as much one is than the other”.  “How many” implies a whole number amount, not a fraction.  Thus 4 x 10^6 is 50 times more than 8 x 10^4 would be a Common Core-aligned answer choice, but choice D is not.

Grade 8

Our final critique is this question, which is unnecessarily wordy.  Why is it necessary to write “right circular cylinder”?  It’s just a cylinder and that’s what the problem should say.  “Cylinder” may have several meanings, but not in Common Core.

There are plenty of related testing issues that we're not touching on in this blog, but at the very least, questions have to be fair.