At the end of May 2013, following its first pilot test for students, Smarter Balanced Assessment Consortium followed up on its previous releases of sample Common Core assessment questions with new batches for grades 3 through 8 and grade 11, in the form of online practice exams.
SBAC has since tweeted (read: crowed) unceasingly about these practice exams...
...and retweeted as well anyone else who’s mentioned the exams...
...in a positive light.
The below embedded quote is attributed to Sue Gendron, policy coordinator of SBAC:
don’t know if this quote describes the previous questions or
the new set. Our email to Gendron, who holds a Ph.D., regarding the accuracy of this
quote, went unanswered. (The New York Times always seems to make repeated attempts to contact people; should we, too, or does that constitute harassment?)
PARCC, whose official sample questions to date can be counted on one’s
fingers, tweeted to us that it plans to release more samples this summer, something akin to a
Friday news dump:
SBAC practice exam has between 21 and 25 questions, and analysis of the questions has already begun. Bowen Kerins, in the blog patternsinpractice, is taking a look at the Grade 11 questions. In the blog mathmistakes, readers commented on a Grade 8 probability question, which we reprint below:
begin our own analysis by affirming that the mathematics behind this
question is sound; simply put, the probability of independent events all
occurring is the product of the probabilities of each event.
The SBAC question’s details and where it’s placed is another matter entirely.
can be solved with a skill not covered in Common Core Grade 8 but in
Grade 7, creating a tree diagram, which would need to be visualized in
the student’s head or drawn on paper. Tree diagrams for outcomes of
“compound events” are found in CCSSI 7.SP.8a, which states, “8.
Find probabilities of compound events using organized lists, tables,
tree diagrams, and simulation. a. Understand that, just as with simple
events, the probability of a compound event is the fraction of outcomes
in the sample space for which the compound event occurs.”
To be complete for this SBAC question, the tree diagram method would require students to draw
210 lines, determine the total number of outcomes (180=5 x 6 x 6), and then
design the spinner in such a way that it creates 18 favorable outcomes,
or 10% of the total.
the problem is far more easily solved by understanding how to calculate
the overall probability of independent events, which is found in
Common Core’s S-CP-2, a high school standard:
that two events A and B are independent if the probability of A and B
occurring together is the product of their probabilities, and use this
characterization to determine if they are independent.”
diagrams, 210 separate lines in this particular problem, are somewhat
unwieldy—an understatement—when problems become complex. The second
method is far easier. So SBAC is attempting to test in Grade 8 a
question that really should assess the understanding and application of a high school standard.
that’s not all. The question’s designers don’t fully align the
question to the high school standard, either. According to the
question, the player wins the game if two outcomes both occur;
presumably meeting the requirements on the spinner is one result and
meeting the requirements on the dice is the other result. However,
these requirements don’t translate cleanly into the actual probability
calculation. That’s because this question doesn’t have just two, but
three independent events. The two dice are independent of each other.
(Is it possible the designers really meant to require “the sum of the pips
on the dice is even”, which would change the game?)
the mathematics needed to find the correct answer, three independent
events, (two red sections = ⅖) x (one die = ½) x (other die = ½) = 1/10
or 10%, goes beyond the scope of Common Core.
All of this analysis so far goes to the mathematics behind the question, but there’s still more:
Joshua Zucker commented on mathmistakes, “Those sure aren’t standard
dice. I wonder how many of their sides are even?” True that. Standard
dice have the opposite sides’ pips add to 7, and on one of these die you can see 3
& 4; the other 2 & 5. But not only are they non-standard dice,
they’re not even the same as each other:
closely. If each die is oriented with the two pip side on top and the
three pip side in front, the left die has the pips closest to each other
on the right side; the right die has the pips closest to each other on
the left. Which means we really shouldn’t be making any assumptions
about the rest of the dice’s sides.
The language in the question? Confusing, ambiguous or omitting essential information, as is often the case.
More common language: number cubes are “dice”. Or they should have numbers on the faces, not pips.
Necessary stated conditions: the dice are “fair”; the spinner is divided into five equal sections.
question could be more succinctly asked: “You’re designing this game.
Drag a color into each section of the spinner so that the probability
of a player winning is 10%.”
By Grade 3, students are supposed to be able to “solve
two-step problems using the four operations. Represent these problems
using equations with a letter standing for the unknown quantity.” See 3.OA.8
Does this SBAC question align to the standard?
possible equation to represent this problem would be 4x + 3 = 27, where
x is defined as the number of chairs. It seems to us that (1) the
second part of 3.OA.8 is putting algebra into Grade 3, and (2) there are
more than two steps involved.
way, the strangeness of asking students to provide the answer by
dragging folding chairs one-at-a-time onto an empty lawn overshadows
both the questionable standard and the questionable alignment.
Next-generation assessments are supposed to be an advancement in
sophistication and use of modern technology, not a sideshow.
question also creates ambiguities, such as whether the 27 guests
includes Mr. Smith, since it’s presumably his backyard. Are the 27
guests all from the four families? That would mean the average family
size is over six. Does Mr. Smith have no chairs at his house? Where’s
the food supposed to go?
Common Core Grade 4, students study equivalent fractions. They also
learn to multiply (whole number) x (fraction), but not (fraction) x
How well will students apply one or the other skill, either of which can be used to solve this problem?
question makes several design choices. In questions of enumeration, or
counting, one doesn’t start at square 1, rather, one must start at
square 0, which in the problem is off the trail where the word Start is.
But finishing means landing on the last, or 24th square, not where the
word Finish is. It’s clear to us what the problem is asking, but in
the details of this problem, the intent is muddied by poor design.
“Jack is ½ finished”, then by counting, he would be on the 12th of 24
squares. (If Jack had 24 cookies, and he was ½ finished, he would have
eaten 12 cookies.) However, if the problem is read literally, Jack
should be standing exactly on the line with 12 squares behind him and 12
squares in front of him.
answer can be found either by finding the equivalent fraction to ½ with
a denominator 24, or 12/24, by multiplying 24 x ½, or by multiplying ½ x
24. (The exam isn’t expecting the last approach, since that is found
in the Grade 5 standards. We don’t think the designer is expecting the
second approach either.)
another unfortunate design weakness, this problem requires students to
count four times: the original count to determine there are 24 squares,
for which there is no shortcut, and for the three answers. There ought
to be a better way to demonstrate fluency with equivalent fractions than
by having students repeatedly demonstrate a kindergarten skill.
there’s that problem with language again. Many weaker readers, even if they get
the right answer for Jack, are going to stumble on the meaning of the
expression “⅓ finished.” “One-third of the way along the trail” is
all of the words and pictures in this problem, and the three bulleted
conditions, it boils down to a simple arithmetic question: Given four numbers,
24, 25, 28 and 29, determine the pair whose sum is evenly divisible by
6. Then find the quotient.
upon a time, that is exactly how the question would have been asked,
but no longer. Now it has to be “engaging”. (Are fourth graders still
interested in building blocks?)
drawback in these interface questions, to repeat ourselves, is that with each element in a
problem’s design, there is room for ambiguity and misleading language,
much of it unnecessary. For starters, the sentence “The table below
shows the number and color of blocks” puts the word “number” before
“color”, but the table reverses the words. Far worse, the first sentence doesn’t state up front that Maya won’t be using all of the blocks, a key
issue to understanding the problem. Better wording might be, “Maya will use
two different colors to build a tower of blocks. She has four colors to
don’t expect Common Core assessments to abandon attempts at posing “real life”
questions and return to questions framed in purely mathematical terms,
but writers take heed: wording and the order of directives matter.
an SBAC-specific interface design decision, many questions have two related
parts separated by a vertical list of draggable numbers in a rectangle.
We don’t know why they chose this quirky design. In this problem, the
quotient (the number of levels) must be chosen from a list of 11
non-consecutive whole numbers. Genuinely odd.
Common Core’s Standards for Mathematical Practice #3 begins: “Construct viable arguments and critique the reasoning of others.”
The SMPs deserve a blog of their own, but #3 stands out from the
others in that it is not purely mathematical. There’s talk among
teachers of bringing the Socratic method into kindergarten classes,
which gives new meaning to “paper chase”.
“Support/don’t support the conjecture” format questions like this one appear repeatedly in the SBAC sample tests. Andrew Stadel reviewed a Grade 6 question of this type as part of a nice assessment comparison done in a video format. Straw poll: should ccssimath.blogspot.com use a video format?
this SBAC question by its wording (and the others, as well) leads students down a mathematically
indefensible path. In mathematics, one does not prove a conjecture by
one or numerous consistent examples, but one can disprove a conjecture
by a single counterexample.
weakness of this particular question is found by considering together
both the conjecture’s absolute nature (“the only way”) and the wording
in part A. Students can write 3/6 (or many other choices) as the answer
to part A, which is greater than 3/7, but it doesn’t “support”
Kendrick’s conjecture in the mathematical sense of being one step on the
way to completing a proof (described in SMP #3 thusly: “mathematically
proficient students...make conjectures and build a logical progression
of statements to explore the truth of their conjectures.”) We shouldn’t be forcing fourth grade students to make or support fallacious arguments without understanding the big picture.
question is simply asking students to “show how it possible to create a
fraction greater than 3/7 by making the denominator less than 7. Also
show how it is possible to create a fraction greater than 3/7 where the
denominator is NOT less than 7.” Students are being asked to provide
examples and counterexamples, but not prove anything. It's clunky and
additionally confusing because Part B is akin to a double negative—as in the above question, too—but correct mathematics.
Common Core is supposed to be increasing rigor, not fostering bad math. Smarter Balanced is, too.