Emily Fitzgerald

April 10, 2016

Diagnostic Interview

Students:

Johnny is a Kindergartener in Richmond City. He typically struggles with the

concept of subtraction. When using manipulatives, the student has a greater

understanding of subtraction. On formal assessments, the student is not proficient in

subtraction. The student is very restless in most academic situations. The student has

trouble sitting still, concentrating, and finishing their work.

Ellie is a Kindergartener in Richmond City. She is very proficient in subtraction.

She strays away from using manipulatives typically to solve any subtraction problem. On

formal assessments, the student performs well and is typically the top of the class.

Overall, the student typically keeps to themselves in class and other social settings. The

student has no behavioral issues.

Understandings:

The main concept that students will be interviewed about is their understanding of

subtraction. Students will be given the opportunity to show how much they know about

subtraction with the use of counters and multiple methods. The students will be able to

use counters, write number sentences, and draw pictures to come to their answers.

SOL K.6 The student will model adding and subtracting whole numbers, using up to

10 concrete objects.)

Student’s Mathematical Knowledge:

Johnny understands the bases of subtraction. He is able to subtract all numbers by

using manipulatives. Johnny does not understand what a number sentence is, but is able

to write the numbers that he used to solve the problem. I have come to the conclusion that

the class, as a whole, has not been introduced to the term “number sentence,” but may

call it something different, by comparing Johnny’s answers with Ellie’s. I also noticed

that Johnny does not have a good sense of using 5 and 10 as benchmarks (Van De Walle

p.152). This was seen when Johnny was not able to rationalize that 5 is half of 10, like

Ellie was able to do. Johnny also was not able to see 5 or 10 as benchmarks, so he had to

solve all of the problems using counters or drawing pictures (Van De Walle p.152).

Ellie understands the bases of subtraction. She was able to subtract all numbers,

sometimes in her head, and sometimes using manipulatives. Ellie does not understand

what a number sentence is, but is able to write the numbers that she used to solve the

problem. I still hold the conclusion that the class, as a whole, has not been introduced to

the term “number sentence,” but may call it something different, by comparing Johnny’s

answers with Ellie’s. Ellie has a great sense of 5 and 10 as benchmarks (Van De Walle

p.152). When I asked her how many counters there were, she was able to tell me that

there were two groups of 5, which means there is one 10. Ellie continued to make these

connections throughout the interview and was able to solve the questions involving 10

fairly quickly.

Recommended Next Steps:

Johnny’s next steps would include completing more subtraction problems that

include 5 and 10 as benchmark numbers. In order for Johnny to begin to complete two-

digit subtraction, he needs to understand 5 and 10 as benchmark numbers (Van De Walle

p.152). The student may benefit by completing problems using 10 frames and counters so

that he can visualize 5 and 10 (Strategies p.1).

Ellie is prepared to move on to double-digit addition and subtraction. She has a

complete sense of 5 and 10 as benchmark numbers. This is important in beginning

double-digit addition. The student may benefit from more practice of subtraction

especially questions that may require more invested reading to help her read/listen

carefully. I believe that the class as a whole will benefit from beginning to call equations

“number sentences.” This will benefit the students as they continue into higher grades

and more intense mathematics.

Reflection:

This experience helped me to work on my individual questioning, and helped me

to see how to appropriately word things so that students can understand. Sometimes, I

find myself asking questions that are way over some students’ heads. I also learned that

sometimes, especially with younger students, I will have to repeat myself multiple times,

and maybe even ask questions multiple ways so that they can understand. I also learned

that not all students will want to use counters, write a number sentence, or write a picture.

I found that I needed to let the students express their thinking the way they want to

express it because that will be the way that they are more comfortable.

This helped me learn a lot about more open-ended questions to promote inquiry. I

learned how to let the students solve the problems how they wanted to, without giving

them any sort of indication as to how they should be solving the problem. This is

important so that students begin to form their own processes and ideas about math, which

is something that we talk about in class when we think about inquiry. I believe that these

students will move on to flourish in inquiry-based mathematics because of their responses

and problem solving skills.

Reference:

Strategies that Develop 10 as a Benchmark Number. (2015). Retrieved April 18, 2016, from https://www.whatihavelearnedteaching.com/strategies-that-develop-10-as-a-benchmark-number/

Van De Walle, J. A., Carp, K. S., & Bay-Williams, J. M. (2004). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Boston: Allyn and Bacon.

Student: Johnny

Student TaskAsk some or all of the questions. Modify the

interview as student thinking dictates. Encourage students to explain their answers.

Interview NotesAllow students to use concrete, pictorial and/or symbolic

representations, and/or verbalized reasoning to support their answer. Record student explanations and actions while assessing. Keep student

work for analysis. Show students 10 counters How many are there? How do you know?

Johnny counted the counters out loud.

The student did not touch them, but counted them out loud while moving his eyes to every counter.

The student came to the conclusion of 10.

The student knew “because I counted them.”

Student TaskAsk some or all of the questions. Modify the

interview as student thinking dictates. Encourage students to explain their answers.

Interview NotesAllow students to use concrete, pictorial and/or symbolic

representations, and/or verbalized reasoning to support their answer. Record student explanations and actions while assessing. Keep student

work for analysis. If you have these counters, and

someone takes away 5, how many do you have left?

How do you know? Can you show me using the

counters?

During interview: “How else could you know, instead of counting?”

Johnny physically moved 5 counters away from the pile.

The student counted the counters that remained.

The student came to the answer “5.”

The student knew “because I counted them.”

Response: “guessing, you can always just guess and hope you get the right answer.”

Student TaskAsk some or all of the questions. Modify the

interview as student thinking dictates. Encourage students to explain their answers.

Interview NotesAllow students to use concrete, pictorial and/or symbolic

representations, and/or verbalized reasoning to support their answer. Record student explanations and actions while assessing. Keep student

work for analysis. If you have these counters and

someone takes away 3, how many do you have left?

How do you know? Can you show me by writing a

number sentence?

Student guesses “4, 3, 2” right after I asked the question without even attempting. Guessed more numbers until I repeated the question.

Student took away 3 counters from the group of 10, counted the remaining counters and came to the answer “7.”

Student was asked to write a number sentence, didn’t understand, so I asked, “Can you write me the numbers that go along with that problem?”

Wrote “7, 10, 3.”

Student TaskAsk some or all of the questions. Modify the

interview as student thinking dictates. Encourage students to explain their answers.

Interview NotesAllow students to use concrete, pictorial and/or symbolic

representations, and/or verbalized reasoning to support their answer. Record student explanations and actions while assessing. Keep student

work for analysis. If Scott has 7 cupcakes, but

someone eats 5, how many cupcakes does he have left?

How do you know? Can you show me by writing a

number sentence? Can you draw a picture?

Student guessed “6.” I asked “are you sure.”

The student then cam to the conclusion of 5, because he assumed I was asking a question about 10 cupcakes not 7.Repeated question

Johnny drew a picture, and was able to come to the conclusion that there were 2 cupcakes left.

“I knew because of the picture I drew.”

Student TaskAsk some or all of the questions. Modify the

interview as student thinking dictates. Encourage students to explain their answers.

Interview NotesAllow students to use concrete, pictorial and/or symbolic

representations, and/or verbalized reasoning to support their answer. Record student explanations and actions while assessing. Keep student

work for analysis. If Jenny has 10 cupcakes, but

someone eats 4, how many cupcakes does she have left?

How do you know? Can you show me by drawing

a picture? Can you write a number sentence?

Student automatically answered “6.”

“It was in my brain, I know that 10 minus 4 is 6.”

Showed me by adding more cupcakes to his picture to make 10, then crossed 4 off to show there were 6 left. He also wrote the numbers “10, 4, and 6, above the picture.

Student : Ellie

Student TaskAsk some or all of the questions. Modify the

interview as student thinking dictates. Encourage students to explain their answers.

Interview NotesAllow students to use concrete, pictorial and/or symbolic representations,

and/or verbalized reasoning to support their answer. Record student explanations and actions while assessing. Keep student work for analysis.

Show students 10 counters How many are there? How do you know?

Ellie counted the counters in her head.

She gave me the answer “10” quickly.

When I asked how she got that, she told me because there are 5 plus 5 and that means there are 10. She pointed out that there are two groups of 5, and “5 is half of 10.”

If you have these counters, and someone takes away 5, how many do you have left?

How do you know? Can you show me using the

counters?

The student immediately answered “5.”

“When you take away 5, there are 5 left.”

“5 is half of 10.”

The student shows me using counters by removing 5 from the pile.

Student TaskAsk some or all of the questions. Modify the

interview as student thinking dictates. Encourage students to explain their answers.

Interview NotesAllow students to use concrete, pictorial and/or symbolic representations,

and/or verbalized reasoning to support their answer. Record student explanations and actions while assessing. Keep student work for analysis.

If you have these counters and someone takes away 3, how many do you have left?

How do you know? Can you show me by

writing a number sentence?

Student showed me by moving 3 counters away and counted what remained. “7.”

Student was asked to write a number sentence, didn’t understand, so I asked, “Can you write me the numbers that go along with that problem?”

Student wrote a 3, 7, and 10 on her paper.

Student TaskAsk some or all of the questions. Modify the

interview as student thinking dictates. Encourage students to explain their answers.

Interview NotesAllow students to use concrete, pictorial and/or symbolic representations,

and/or verbalized reasoning to support their answer. Record student explanations and actions while assessing. Keep student work for analysis.

If Scott has 7 cupcakes, but someone eats 5, how many cupcakes does he have left?

How do you know? Can you show me by

writing a number sentence? Can you draw a picture?

Ellie counted 7 counters out, then moved away 5 from the pile.

The student came to the conclusion of “2.”

The student was unable to draw me a picture of what this may look like. The student was able to write me a “2” on her paper.

If Jenny has 10 cupcakes, but someone eats 4, how many cupcakes does she have left?

How do you know? Can you show me by

drawing a picture? Can you write a number sentence?

Student automatically answered “3”

Student did not listen to the question correctly, thought I was asking 7-4, not 10-4.

Used 7 counters and moved away 3.

Student did not catch the mistake when I re-read the problem.

Ellie could not write a picture or write a number sentence, but she wrote “3” on her paper.