Study Guide
Field 212: MultiSubject: Teachers of Early Childhood
(Birth–Grade 2)
Part Two: Mathematics
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Sample ConstructedResponse Item
Competency 0005
Analysis, Synthesis, and Application
start bold Use the data provided to complete the task that follows. end bold
Using the data provided, prepare a response of approximately 400 to 600 words in which you:
 identify a significant mathematical strength related to the given standard that is demonstrated by the student, citing specific evidence from the exhibits to support your assessment;
 identify a significant area of need related to the given standard that is demonstrated by the student, citing specific evidence from the exhibits to support your assessment; and
 describe an instructional intervention that builds on the student's strengths and that would help the student improve in the identified area of need. Include a strategy for helping the student build a viable argument related to the given standard.
Background Information
Firstgrade students have been developing an understanding of adding two singledigit numbers and representing the process as an equation. The students have been adding and subtracting within 20 using strategies such as counting on, making ten, and decomposing a number leading to a ten. The class is currently working on the following standard from the New York State P to 12 Common Core Learning Standards for Mathematics.
start bold Operations & Algebraic Thinking (1.OA) end bold
start bold Work with addition and subtraction equations. end bold
7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.
The teacher has planned a lesson experience in which students will use number cards and recording sheets. The teacher has the students work in groups of two.
Description of Class Activity
In the teacher's lesson experience, pairs of students practice combining two numbers while playing a game called "Capture 4." The teacher gives each pair of students a deck of illustrated number cards and recording sheets for writing equations involving addition. The cards are numbered from 0 to 9. To play the game, each student turns over two number cards, finds the sum of the two numbers on the cards, and then writes an equation for the addition problem on the recording sheet. The player with the largest sum "captures" all four cards. Two cards and a completed recording sheet are shown in the example below.
Capture 4 Recording Sheet
Name: StudentWrite your equation below. 8 plus 6 equals 14
Excerpt of Group Discussion
As students work, the teacher moves among them to observe their activity and ask questions. The teacher notices that students use a variety of adding strategies. For example, some students count using the dots on both cards, and others start with the total on one card and then count up using the dots on the other card. The teacher stops to observe one group's work in progress and asks the students to explain their strategies for computation. The group's work is shown below, accompanied by an excerpt of a discussion between the two partners and the teacher.
Kaleme and Isabella have turned over the following cards.
Kaleme has two cards, one with seven dots and one with eight. Isabella also has two cards, one with six dots and one with five.
Kaleme: I won! Isabella: Wait, that's too fast. How did you add them up so fast? Kaleme: I didn't have to add. My cards have more dots than yours, so I must have more. Isabella: That's not fair. You still have to add them up and write the equations down. Then it's my turn, and then we decide who won. The teacher observes as Kaleme covers two of the dots on the 7 card with his finger and counts the remaining dots.
Kaleme: (counts out loud) 1, 2, 3, 4, 5. I have 15. The teacher decides to intervene and ask Kaleme some questions.
Teacher: Can you explain how you got your total, Kaleme? Kaleme: I knew that I needed 2 more to make 10 on the 8 card. So, I took 2 off of the 7 card (illustrates this by covering two of the dots again). Then I just counted what's left on the 7 card: 1, 2, 3, 4, 5. So, 10 with 5 more is 15. So, now I can record it. Kaleme fills out his recording sheet as shown below.
Capture 4 Recording Sheet
Name: KalemeWrite your equation below. 8 plus 2 equals 10 plus 5 equals 15
Teacher: I agree with the sum that you found but let's look closer at the equation. Kaleme: I think the equation is right because it is true that 8 plus 2 equals 10 and 10 plus 5 equals 15 . Those are easy sums to figure out.
Sample Strong Response to the ConstructedResponse Assignment
Kaleme demonstrates a significant mathematical strength in his ability to decompose
a number leading to a ten. With this knowledge he appears to be able to apply the
associative property to draw his conclusion that his two cards have more dots than
his partner's. Knowing 2 more than 8 was needed to make 10, Kaleme decomposed 7 to
Kaleme demonstrated a significant area of need in his misuse of the equality symbol, using " the equal sign " as an arrow leading to an answer to a computation, rather than as a symbol meaning that the expressions on either side of it must have the same value. After writing " 8 plus 2 equals 10 plus 5 equals 15 " Kaleme says it is true because " 8 plus 2 equals 10 and 10 plus 5 equals 15.
Instructional intervention should build on Kaleme's understanding of the associative
property and help clarify the meaning of the equality symbol. Asking "What does
= mean?" would be a first step toward assessing what the student is thinking. The
teacher would then work with Kaleme on his erroneous equation and discuss what is
right with it. Kaleme sees that
8 plus 2 equals 10
10 plus 5 equals 15
Following the logical progression of his own statements and then analyzing the flawed logic resulting from the misused equality sign when it is pointed out to him will help Kaleme build a viable argument.
Kaleme should practice this by playing a matching game that includes cards with
equations on them such as
4 plus 2,
and
3 plus 3,
etc., and cards with an equal sign. He would make equations, illustrating that what was
on one side of the equal sign was, in fact, equal to what was on the other side:
3 plus 3 equals 4 plus 2,
3 plus 7 equals 10
10 plus 6 equals 16
Throughout the activity, the teacher should continue to ask the student to explain the meaning of the equals symbol and describe what he is doing.
Performance Characteristics for ConstructedResponse Item
The following characteristics guide the scoring of responses to the constructedresponse assignment.
Completeness  The degree to which the response addresses all parts of the assignment 

Accuracy  The degree to which the response demonstrates the relevant knowledge and skills accurately and effectively 
Depth of Support  The degree to which the response provides appropriate examples and details that demonstrate sound reasoning 
Score Scale for ConstructedResponse Item
A score will be assigned to the response to the constructedresponse item according to the following score scale.
Score Point  Score Point Description 

4 
The "4" response reflects a thorough command of the relevant knowledge and skills:

3 
The "3" response reflects a general command of the relevant knowledge and skills:

2 
The "2" response reflects a partial command of the relevant knowledge and skills:

1 
The "1" response reflects little or no command of the relevant knowledge and skills:

U  The response is unscorable because it is unrelated to the assigned topic or off task, unreadable, written in a language other than English or contains an insufficient amount of original work to score. 
B  No response. 