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Study Guide

Field 212: Multi-Subject: Teachers of Early Childhood
(Birth–Grade 2)
Part Two: Mathematics

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Sample Constructed-Response Item 1

Competency 0005 
Analysis, Synthesis, and Application

start bold Use the information provided in the exhibits to complete the assignment that follows.end bold

Using the data provided, prepare a response of approximately  400 to 600  words in which you:

Background Information

First-grade students have been developing an understanding of adding two single-digit 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 through 12  Learning Standards for Mathematics.1

 start bold Operations & Algebraic Thinking (UCARTAP NY-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.

Two cards, one with eight dots and the other with six.

Capture 4 Recording Sheet

Name: Student
Write 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: Kaleme
Write 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 Constructed-Response Item 1

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  2 plus 5,  as shown by covering up two dots on the 7 card and counting the remaining dots. To explain how to get the total, his response was, "I took 2 off the 7 card. Then I just counted what's left on the 7 card: 1, 2, 3, 4, 5. So, 10 with 5 more is 15." The student appears to understand the associative property since he considered  8 plus 7  or  8 plus open parenthesis 2 plus 5 close parentheses  to be the same as  open parentheses 8 plus 2 close parentheses plus 5  or  10 plus 5. 

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  and that  10 plus 5 equals 15  , which shows that he does have an understanding of equality. The teacher would then break apart the equation so that he can see where the confusion arises and where he has made a mistake in writing the equation. The teacher would cover up the " plus 5 equals 15 " portion and ask him if the equation showing is correct. She would then do the same with showing only " 10 plus 5 equals 15 " and he will see that this is also correct. Finally, the teacher would cover up the " equals 15 " and ask if the remaining equation is correct—which, of course, it is not. She would then ask him what he thinks went wrong and lead him to the discovery that he needed to write the equation as individual steps:

 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,   4 plus 2 equals 6,  etc. It would also be helpful to have him write the equations as illustrated above, as he continues to play the Capture 4 game with his partner as this will require writing a few equations to come up with his final answer: for example,

 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.

Sample Constructed-Response Item 2

Competency 0005 
Analysis, Synthesis, and Application

start bold Use the information provided in the exhibits to complete the assignment that follows. end bold

Using the data provided, prepare a response of approximately  400 to 600  words in which you:

Background Information

Prekindergarten children have been developing an understanding of the different kinds of shapes with straight sides, including squares, rectangles, and triangles. The class is currently working on the following standard from the New York State Pthrough12 Learning Standards for Mathematics.2

 start bold  Geometry (NY-PK.G)  end bold 

 start bold  Identify and describe shapes (squares, circles, triangles, and rectangles).  end bold 

2. Name shapes regardless of size.

Description of Class Activity

The teacher has set up an interactive lesson experience during which children will use sticks and straws to create squares, triangles, and rectangles and name each shape. The children will work in groups of two. The teacher walks throughout the room to observe the way the children interact with one another and create models, gathering information to determine whether the class is ready to move to a new topic or identify any common misconceptions that need to be addressed.

Excerpt of Group Discussion

Teacher: Wow, you have been busy! Can you tell me what you've made?

One of the students, Kinsey, points to her four shapes, which are shown below.

a row of 4 shapes An image of four shapes arranged in a line is shown. The first two shapes are squares and the last two shapes are equilateral triangles. The left square is much larger than the other shapes. The left equilateral triangle has a vertex pointing up, the right equilateral triangle has a vertex pointing down.

Kinsey: I made four shapes.
Teacher: Can you tell me what they are?
Kinsey: This is a square (points to the small square). This is a rectangle (points to the big square). This is a triangle (points to the small triangle). This is a different triangle (points to the upside-down triangle).
Teacher: (points to the first two shapes) How did you know to call these a square and a rectangle?
Kinsey: Squares and rectangles have four sides. These shapes have four sides. And, they look like boxes. Rectangles are bigger than squares. The bigger one with four sides is a rectangle. The little one with four sides is a square.
Teacher: How did you know to call the shapes "triangles"?
Kinsey: Triangles only have three sides. These are different because one is pointing up, and one is upside down.

Sample Strong Response to Constructed-Response Item 2

The student shows a significant mathematical strength in that she can recognize the difference between a three-sided shape and a four-sided shape, identifying two squares as shapes with four sides and the two triangles as shapes with three sides. She notes that the shapes she identifies as square and rectangle each have four sides and look like boxes, relating their shape to something she is familiar with - a box. While she refers to one of the triangles as different because it is upside down, she still sees it as a triangle because it has three sides. The orientation of the triangle does not affect her understanding of them both as triangles. She also understands the names of the four different shapes she has made in relation to the shapes her class is learning about, and she is partially correct in her description of a rectangle.

Kinsey has a significant mathematical area of need when it comes to recognizing the difference between a square and a rectangle. While she understands that both shapes have four sides, she believes that the difference between a rectangle and a square is that a rectangle is bigger, and therefore calls the bigger square she has made a rectangle. This may be because she has been told that a rectangle has four sides, with two sides bigger than the other two sides, but it has caused a misunderstanding that a rectangle is simply bigger.

As an intervention, I would start by having Kinsey tell me again what the similarities are between the two square shapes she has made. She will repeat that they both have four sides, which she has already identified. I would then review with her that a rectangle has four sides, but that with a rectangle two sides are the same length as each other as are the other two sides, but that one set of sides is longer - or bigger - than the other set of sides. I would then make a rectangle, using other straws, to illustrate what this looks like, and I will have her note that there are two shorter sides and two longer - or as she has called them, bigger - sides. I will then point to different rectangles nearby - a piece of paper for example - and ask her what the shape is and have her explain how she can tell the shape is a rectangle (four sides, with two sides longer than the other two sides). From here, I will work with Kinsey and her partner to make several four-sided shapes having them identify whether the shape is a square or a rectangle. I will want her to be able to show me that the two sets of sides are different lengths in a rectangle, but they are all the same length in a square. As a follow-up activity, I will have Kinsey find objects in the room that are either rectangles or squares, and have her describe which shape is which, while explaining how she can tell the difference between the two.

Performance Characteristics for Constructed-Response Item

The following characteristics guide the scoring of the response to a constructed-response item.

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 Constructed-Response Item

A score will be assigned to the response to a constructed-response 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:
  • The response thoroughly addresses all parts of the assignment.
  • The response demonstrates the relevant knowledge and skills with thorough accuracy and effectiveness.
  • The response is well supported by relevant examples and details and thoroughly demonstrates sound reasoning.
3 The "3" response reflects a general command of the relevant knowledge and skills:
  • The response generally addresses all parts of the assignment.
  • The response demonstrates the relevant knowledge and skills with general accuracy and effectiveness.
  • The response is generally supported by some examples and/or details and generally demonstrates sound reasoning.
2 The "2" response reflects a partial command of the relevant knowledge and skills:
  • The response addresses all parts of the assignment, but most only partially; or some parts are not addressed at all.
  • The response demonstrates the relevant knowledge and skills with partial accuracy and effectiveness.
  • The response is partially supported by some examples and/or details or demonstrates flawed reasoning.
1 The "1" response reflects little or no command of the relevant knowledge and skills:
  • The response minimally addresses the assignment.
  • The response demonstrates the relevant knowledge and skills with minimum accuracy and effectiveness.
  • The response is minimally supported or demonstrates significantly flawed reasoning.
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.

Acknowledgments

1From the New York State Education Department. New York State Next Generation Mathematics Learning Standards. Internet. Available from http://www.nysed.gov/curriculum-instruction/new-york-state-next-generation-mathematics-learning-standards; accessed 6/4/2020.

2Ibid.