Addition problem: James wants to paint racing stripes around his room. His room is 11 feet long and 10 feet wide. If James paints a stripe to go around the top and bottom of his room, how many linear feet of racing stripes will he need?
We might represent the reasoning involved in this problem in the following way.
♦ 10 + 10 + 11 + 11 = 42 feet
♦ Top of room = 42 feet
♦ Bottom of room = 42 feet
♦ 42 + 42 = 84 feet
James will need to paint 84 feet of racing stripes.
2. Subtraction problem: Kelly works at her family’s pet store. She put 272 bags of dog food on the shelf. Last week, customers bought 117 bags. How many bags were left? Her parents also need to place an order to buy more dog food when they reach 50 bags. How many more bags can they sell before they need to place a new order?
We might represent the reasoning involved in this problem in the following way.
♦ 272 – 117 = 155 bags remaining
♦ 155 – 50 = ?
♦ 155 – 50 = 105
The family can sell 105 bags before they need to reorder.
3. Multiplication problem: Aidan collects baseball cards. His collection currently includes 48 players. His brother has 4 times as many cards, and his friend has 3 times as many. How many cards do Aidan’s brother and friend have in all?
We might represent the reasoning involved in this problem in the following way.
♦ 48 × 4 = brother
♦ 48 × 3 = friend
♦ Brother = 192 cards
♦ Friend = 144 cards
♦ 192 + 144 = 336 cards
Aidan’s brother and friend have 336 cards in all.
4. Division problem: Libby collects concert shirts. Each shirt costs $12. If Libby has $120, how many concert shirts can she buy? If Libby saved $9 per month, how long would it take her to save enough money to buy that many shirts?
We might represent the reasoning involved in this problem in the following way.
♦ $120 ÷ 12 = 10 shirts
♦ Libby can buy 10 shirts with $120.
♦ $120 ÷ $9 = 13.33
It would take Libby thirteen months and about ten days to save enough money to buy ten shirts.
Clearly the steps in reasoning necessary to solve these problems have some significant differences from type to type. The steps to solving the addition problem are straightforward. Students find the perimeter by adding the length of the sides and then doubling that quantity to account for the two stripes.
The steps to the subtraction problem are more involved. Students first determine the remaining number of bags after 117 are sold. This is quite simple. Students then compute the difference between the remaining number of bags and the threshold number of 50. Although this step involves subtraction, as did the first step, it has a totally new perspective.
The multiplication problem begins with students multiplying two numbers—one for the brother and one for the friend. The next step involves addition.
The division problem involves the most complex set of steps. It begins with a straightforward division task—the total amount of money available divided by the cost of each shirt. The problem then shifts contexts. Students must take the total amount of money available and divide it by an amount of money that Libby can save each month. However, this step also involves dealing with a remainder, which adds complexity.
Looking at the problem types, the subtraction problem is the only one of the four that requires students to perform an operation on a quantity that the directions to the problem do not explicitly state. The division problem is the only one that involves a remainder. One can also make the case that each of the four problems makes some unique cognitive demands simply to understand it.
The reason this standard is equivocal, then, is that it does not make clear how important the four operations are to demonstrating proficiency. It does not make clear how important remainders are to demonstrating proficiency, and it seems to treat four operations equally, although they have significant differences in their execution.
As another example of equivocality in standards, consider the following high school ELA standard and the upper elementary civics standards.
ELA (high school): Analyze multiple interpretations of a story, drama, or poem (e.g., recorded or live production of a play or recorded novel or poetry), evaluating how each version interprets the source text (RL.11–12.7; NGA & CCSSO, 2010a)
Civics (grades 3–5): Identify the major duties, powers, privileges, and limitations of a position of leadership (e.g., class president, mayor, state senator, tribal chairperson, president of the United States) … evaluate the strengths and weaknesses of candidates in terms of the qualifications of a particular leadership role. (section H, standard 1; Center for Civic Education, 2014)
In both of these examples, assessments would be quite different depending on a teacher’s selection of available options. For example, in the ELA standard, comparing the treatment of the same content in a story and a poem is a quite different task from comparing a story and a play. In the civics standard, knowing the duties, powers, and privileges of a class president is a quite different task from knowing the duties of a state senator.
Standards as Inconsequential
We have observed two practices that appear to address the problems associated with standards but, in fact, render standards inconsequential: (1) tagging multiple standards and (2) relying on sampling across standards.
Tagging Multiple Standards
One common practice is for teachers to assess standards by simply tagging multiple standards in the tests they give. For example, assume that a teacher has created the following assessment in a seventh-grade ELA class.
We have been reading Roll of Thunder, Hear My Cry by Mildred D. Taylor, which tells the story of Cassie Logan and her family who live in rural Mississippi. In the novel, Taylor develops several themes. Describe how the author develops the theme of the importance of family through characters, setting, and plot. Compare the importance of family theme with one other theme from the book. Write a short essay that explains which of the two themes you think is the most important to the development of the novel. Justify your choice with logical reasoning and provide textual evidence.
Because the teacher must cover all the seventh-grade ELA standards, he or she simply identifies all those standards directly or tangentially associated with this assessment. For example, the teacher might assert that this assessment addresses the following Common Core standards to one degree or another:
RL.7.1
Cite several pieces of textual evidence to support analysis of what the text says explicitly
