Amber Hornbarger

Fourth Grade Diagnostic InterviewMontrose Elementary

Henrico County

Amber Hornbarger

April 14, 2016

hornbargerar@vcu.edu

Late because of MAPS testing.

Amber Hornbarger

Students

When I asked my fourth grade teacher for two students to interview about comparing and

ordering fractions, she offered me a high achieving student and a low achieving student. Both

students are well behaved students that are outspoken in class. Kiara is the high achieving math

student. The worksheet below is an example of her work. Kiara has strong number sense and

understands how decimals, word names, and fractions relate to one another. She has a strong

understanding of what a fraction is and what it means to have part of a whole.

Rayne on the other hand is a low achieving math student. She does very well in English,

Reading, and Social Studies. When I taught my Social Studies lesson she was on a roll answering

every question correctly. In math, she is less confident in herself. She does not have strong

number sense and earned a “C” on the last math checkpoint quiz. She seems to get ahead of

herself and move too quickly from what I have observed in class.

Amber Hornbarger

Interview Protocol

Virginia Math Standard 4.2 The student will a) compare and order fractions and mixed numbers; b) represent equivalent fractions;

In association with SOL 4.2, I wanted to explore student understanding of fractions,

mixed numbers, and equivalent fractions. Students should be able to recognize fractions and

mixed numbers as they have been working on constructing the idea that whole units can be

broken into equal parts and how to use models to judge size of fractions. They also have been

working on recognizing and generating equivalent forms of fractions. I will assess their

knowledge of the above and evaluate their comfortability using more than one method to solve

problems that involve fractions. I will also assess student understanding of the fact that the more

parts a whole is broken into, the smaller the parts.

Amber Hornbarger

Student Task

Ask some or all of the questions. Modify the interview as

student thinking dictates. Encourage students to explain their

answers.

Interview Notes

Allow 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.

35 >

38

Is the above equation true?

How do you know?

Symbolic / Verbal

Using manipulatives, can you please order these

fractions?

25

310

12

45

38

810

Concrete / Verbal

Draw a picture to answer this problem.

Which fraction is larger?

45

57

Pictoral / Symbolic

Using decimals, solve this problem.

29+ 7

12=¿

Symbolic

Please write two sentences explaining what a fraction is.

Please write one sentence explaining what a mixed

number is.

Answer this question: The more parts the whole is divided

into, the _________ the parts. Smaller or larger.

Written

Amber Hornbarger

Interview 1 – Kiara Kiara read the problem and immediately turned to her scratch paper and rewrote the fraction. She cross multiplied the fraction and then told me that the equation was true because she cross multiplied. I asked her to write that down for me.

Then I asked Kiara if she couldn’t multiply, would she know which one was bigger? She stared at it for a few minutes and then looked at me and asked if she could move on to the next problem.

Kiara started by laying out 2/5 and 3/10. She then doubled

the 2/5 line using same colors and combined colors on the

line. So took the 2 yellows and 5 blacks from each and made

one long line. Then she put a white dot in the middle. I

asked her why she didn’t do ½ and she said that she didn’t

have to because she knows it belongs in the middle. Then

she laid out 3/8 and 3/10 and she said 10 has smaller parts so

8 is bigger. She would not elaborate on why 3/8 is smaller

than 4/10. The picture above depicts her order with a dot in

the middle for ½. She did this with ease. I asked her how she knew where each line went. She

said that 2/5 is equal to 4/10s because she doubled the lines so she did the same with 4/5 which is

equal to 8/10 so they belong at the end.

Amber Hornbarger

For the third question, Kiara drew the picture and then looked

at me like what in the world is this. I asked her which fraction

was larger and she said “I have no idea, they are too close.” So

I asked her if there were any other ways that she could compare

the fractions. She said “cross multiply!” and proceeded to do

just that. Then she circled 4/5 which is the correct answer. She

said 28 is bigger than 25 after cross multiplying and that is how

she could check her work.

For the fourth problem, I asked Kiara to use decimals to solve

the problem. Instead she created equivalent fractions by

increasing fraction sizes in increments. When she reached a

common denominator she added them together, coming to 29 / 36ths. I then asked her if she

knew what 29/36ths was in decimal form. She responded with “If I had my calculator I’d tell you

to divide them”.

Kiara answered the last problem on her own without assistance.

Amber Hornbarger

Interview 2 – RayneRayne answered the first problem without any assistance from me. I asked her how she knew to multiply and she said “because that tells you which one is bigger”. I asked her if she knew why it worked that way and she shrugged.

With the manipulatives, Rayne did something that I did not expect and she looked at the denominators and decided that 8 was the only unlike number so she put it at the top. She turned all of the 5s into 10s. She told me as she did it that she knew every fraction just had to be doubled because it was the same as the 6/12 problem they had yesterday. Her results are above.

Amber Hornbarger

On the third problem, this is the picture that Rayne drew. She said that 4/5 seemed bigger because it took up more room in the one sentence than 5/7s did. I asked her if there was another way she could check her guess. She cross multiplied and stuck with her original answer which was a correct guess.

Rayne asked if she could skip the fourth problem. I didn’t push her.

Amber Hornbarger

Rayne answered the fraction portion of the last problem without assistance. When she read mixed number she asked me “Is that the one that’s like 1 and 1/4th”. I said yes and that is the sentence she wrote post validation.

Amber Hornbarger

Evaluation of Interviews

Both students have a strong understanding that fractions are a part of a whole. They both

understood how to represent fractions concretely even if they did not order them correctly. Both

students understood common denominators and how to compare fractions with common

denominators. Both students knew how to find equivalent fractions but were uncomfortable

doing so unless it was in multiples of a half or a benchmark number. They did not use a LCM

method. They more or less chose the next number that they thought of and they had benchmark

numbers to use so I did not witness critical thinking in regards to equivalencies. According to the

textbook on page 341, “students build on their prior knowledge, meaning that when they

encounter situations with fractions, they naturally use what they know about whole numbers to

solve the problems.” I believe the students do not fully understand previous concepts such as

LCM and or GCF that could greatly aid them in the journey of fractions.

They both knew that a mixed number had two parts and that decimals represented the

same relationship as a fraction. Both students only knew and or felt comfortable using one

method (cross multiplying) to compare fractions.

Kiara truly understood that the more parts that a whole was divided into, the

smaller the parts. Rayne could define it but did not apply it.

Kiara had strong number sense and understood how decimals, percentages, and

fractions relate. Rayne did not display an understanding of this concept.

Kiara understood how to compare and order fractions using common

denominators. Rayne knew how to use common denominators for benchmark

numbers but was not able to apply knowledge of smaller pieces with the 3/8 vs

3/10.

Amber Hornbarger

Next Steps

Both students would benefit from deeper teaching in regards to various methods to reach

the same answer. They know one method of comparing but I believe they need to see the concept

in various forms of visuals and hands on materials so that they can apply the definitions that they

have memorized. I believe that deeper studies into the various methods would also build on the

number relations knowledge they need to understand how parts relate to the whole and how

pieces of a pie are divided into a fraction. In other words, I believe they need more experience

with manipulating fractions in authentic ways so that they can understand breaking something

from ½ into 2/4ths and then into 4/8ths can create smaller pieces of the pie.

For Kiara, I believe that she generally needs further experience with fractions as stated

above but I also think that she could benefit from further exploration within larger numbers to

see if the concepts learned really stick. I believe that Kiara could move forward with adding and

subtracting fractions. I also believe that she could move forward with SOL 4.5, solve single-step

and multistep practical problems involving addition and subtraction with fractions and with

decimals.

For Rayne, I would use the advice from page 341 in the textbook in which it addresses

the misconception that a fraction such as 1/5 is smaller than a fraction such as 3/10. The book

tells me to use various visuals and contexts to show parts to wholes. Use hours of time or pizza

slices in regards to sharing. I would use that as my next step to help her understand fraction sizes

and in turn further strengthen her ordering and comparing skills. Rayne also needs to further

explore the relations between ratios, division, percentages, decimals, and fractions.

Amber Hornbarger

ReflectionThis experience involved what I would consider to be key elements of this course by

demonstrating that students that are only taught through direct instruction without much

inquiring do not develop the number relationships understandings to nurture deeper

understandings. It also gave me the opportunity to practice guiding questions and dig deeper into

student thinking. I was able to truly analyze and examine student thoughts and use that

examination to probe further and further until I was able to determine which portion of the

concept the student did not understand. Hand in hand to what the key elements of the course are,

I learned a lot from this experience because it was the first time that I dug this far in depth into

student work to understand how they are developing their math understandings. It made me

really think about how the students perceived my interview questions and how my verbal

questions can guide students to think one way or another. Then that allowed me to gather enough

information to create a second lesson that can teach the students what they are missing in their

current understanding. That was probably the most beneficial piece of experience for me because

I do not have much experience with probing students and guiding students through inquiry. I

finally had the opportunity to put the textbook into practice in such a small scale that I was able

to truly focus in on student engagement and understanding.