Diagnostic Interview
Syllabus Description
The purpose of this assignment is to develop your skill in assessing students’ understanding of mathematics
concepts. You will be required to design and conduct diagnostic interviews to better understand students’
mathematical need(s) that became apparent to you during your Practicum B experience. These individual interviews
will take place with three different students on one specific mathematics concept (i.e., design one flexible interview,
conduct it individually with three students). This assignment has the following five parts:
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5.
Choose and describe the students you will assess;
Design an interview protocol to assess the students’ knowledge of the target mathematics concept;
Conduct the interview assessment with the students;
Evaluate the students’ mathematical knowledge; and
Recommend next steps for learning experiences for these students.
Please see the rubric for specific details within each category. Organize the report using the five sections, synthesizing
your responses as you see fit to coherently address the rubric. For example, you will need three descriptions (one per
student) in Part 1, but only one flexible interview protocol in Part 2. Part 3 will require submitting three sets of interview
notes, while Part 4 should clearly communicate your interpretation of each child’s understandings, including some
comparison. Part 5 should include both individual and group recommendations. Please refer to Blackboard and rubrics
for specific details of the assignment.
This assignment addresses the goals of becoming using assessment techniques that inform the teacher’s instructional
decisions (CF II.2, ACEI 4). It addresses the Course Objective of Understanding Children’s Thinking.
Diagnostic Interview
Rubric
Describe the students you plan to interview (x/30)
This section should consist of three one-paragraph descriptions using pseudonyms. Each paragraph should meet the
following requirements:
 Describe the student’s level of understanding of the target mathematics concept(s) and other mathematics
performance prior to the assessment. Refer to evidence such as informal observations, formal assessments,
classwork, etc. (3-5 sentences)
 Describe the student’s performance in other relevant academic, social, or behavioral areas. (1-2 sentences)
 All interviewed students are in 4th and 5thgrade in Hanover County. The two fourth
graders take math with a mixed group of on-level to above grade level students
from two fourth grade classes. The other class contains students who are above
grade level and passed testing for an accelerated math program of 4th and 5th grade
math within their 4th grade year. The 5th grade student is in a standard math class
with her assigned teacher.
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Hallee is a low achieving math student in 4th grade. Hallee received a B in math on her
report card. Her teacher commented that she has a tough time subtracting larger numbers
when regrouping was involved especially when there were 0's in the top number. Hallee also
doesn't have a well developed number sense.
Maddux is a high-achieving 4th grade math student. He always seems to be the one
with the right answer in math class, constantly raising his hand and commenting to
classmates “this is so easy”. Maddux, like about half of the students in the class,
often works ahead of the teacher as she explains concepts on classwork. Maddux
performs wells on his tests and quizzes. He received an A in math on his report card,
understands concepts the first time they are taught, has a well developed number sense,
loves to play math games and share his understandings with the class.
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Kaylee is an average 5th grade math student. Kaylee received a B grade in math on
her last report card. At the very beginning of the year and in previous years Kaylee
was provided with pull-out services for math but they were deemed unnecessary
shortly after this school year started.
Design a flexible interview protocol to assess the students’ knowledge of a mathematics concept (x/50)
This section should consist of one flexible interview protocol using the format provided in class. Tasks may
contribute to meeting more than one requirement, such as concrete, pictorial, and connected representations.
 Select and clearly describe the mathematics concept (not procedure) for the interview assessment, citing
appropriate standards. (3-5 sentences, plus copy/paste the standards)
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In conjunction with VA SOL 4.15 and 5.17 students should be able to recognize,
understand, reproduce, create and extend geometric and numerical patterns using
concrete materials, number lines, tables and words. Students have been working on
recognizing and extending numerical patterns. They will also be able to extend
growing numerical patterns and learn to generalize. In this interview I will assess
their knowledge of these and push them further to use alternative methods and
representations to demonstrate their understandings.
VA SOL 4.15 The student will recognize, create, and extend numerical and
geometric patterns.
Diagnostic Interview
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VA SOL 5.17 The student will describe the relationship found in a number pattern
and express the relationship.
Include one unused copy of the interview protocol.
Several tasks use concrete representations (manipulatives, measuring tools, or other objects the child can
manipulate during the assessment).
Several tasks use pictorial representations (drawings, diagrams, charts, or graphs that are drawn by the child
or are provided for the child to read and interpret).
Several tasks use symbolic representations (numbers/letters the child writes/interprets to demonstrate
understanding).
Several tasks involve connected representations.
Tasks/questions require the student to explain his/her thinking.
Conduct an interview assessment with three students (x/30)
This section should consist of three sets of annotated interviews. Each annotated interview should meet the
following requirements:
 Include visual examples of anecdotal notes taken during the interview assessment (i.e., scan of the actual
interview page, including en vivo notes).
 Include visual examples of student responses (i.e., images of student work).
 Use Comments or other annotations to add just enough detail to your notes for the reader to “be there”
o Describe what the student did for each task in your interview protocol (e.g., interchangeably used tens
and ones blocks)
o Describe adaptations you made during the process (e.g., started on task 2 because XYZ; moved to task
5 because ABC)
Evaluate students’ mathematical knowledge (x/50)
This section should thoroughly cover all three students and consist of interpretation and comparison/contrast,
organized as best fits your experience.
 Based on the students’ responses in the interview, interpret what the students understand and do not understand.
 By getting to work with a low fourth grader, a high fourth grader and an average fifth
grader from another class, I was able to get a unique perspective on numerical pattern
knowledge across the board. Each student had different strengths and weaknesses that
were surprising to me and all could deepen their mathematical understanding on the topic
in one way or another.
 Hallee’s lacking number sense showed through in her diagnostic interview. Many times I felt
that she could have been successful if she had the understanding of relationships between
numbers to realize that her answer did not make sense. For example, when asked to extend
the second manipulative pattern going to the left rather than the right, she did not realize
that she needed to change the rule from subtraction to addition. She added onto the
pattern in this way and did not recognize that visually the pattern looked wrong since it
went from a lesser number to a high number then back down to a lesser number. Maddux
also lacked this understanding in his interview when he was asked to extend his own
pattern in both directions. This was one of the key understandings that set apart the 4 th
graders from the 5th grader. Kaylee was easily able to tell me that a pattern was continuous
and could be extended in each direction and knew to change the operation of the rule to
continue on in the opposite direction. In Elementary and Middle School Mathematics
Teaching Developmentally by Van De Walle, Karp and Bay-Williams, this understanding is
recommended as the first teaching when introducing patterns to students (p 273).
 Hallee also relied heavily on the manipulative, counting on them during our first task, and
used her fingers when doing mental math. Kaylee did not count on the manipulatives but
rather used mental math during the first task. Later Kaylee also used her fingers while
Diagnostic Interview
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doing mental and symbolic math. Hallee also used the standard algorithm when the problem
was too high for her to use mental math. Kaylee used a non standard algorithm, adding tens
and ones separately when the mental math was too much for her. Maddux used almost
solely mental math and often showed quick thinking while doing it. Much of the symbolic
representation on his work pages were what he wrote while explaining to me what he did in
his head. Hallee and Kaylee’s reliance on maniplualtives and fingers shows a weakness in
quick recall of math facts. As mentioned in the beginning, Hallee’s teacher had noted this
weakness. Kaylee’s weakness in this area may have contributed to her math pull out services
before. Maddux showed strength in his quick recall of math facts and computation but
showed other areas of weakness in mathematical understanding. According to the text,
students should analyze patterns with “developmental progression of reasoning by looking at
the visuals, then reasoning about the numerical relationships and then extending to the
larger nth case” (p 274).
One other missing concept shown in Hallee’s interview that contributed to many of her
wrong answers, specifically in the first and third tasks, was that she assumed that is a rule
was true for two numbers in the pattern than it must be true for all. This understanding is
okay as a beginning concept of patterns but should be extended to double rule (recursive)
and growing patterns in which you have to find the relationship between multiple numbers
before deciding the rule of the pattern, Because of this in task three she missed the -13
part of the rule and came out with an incorrect answer.
I felt that Maddux’s interview was the most interesting of the three. I have always had an
interest in gifted and talented students (his teacher did not mention if he has been labeled
as “GT” in his classroom, but from what I’ve seen of his thinking I’d assume he was if not,
borderline) and the incredible ways that their brain works. But as if often the problem with
these types of students, their brain can work so quickly to find the answer that when asked
to think outside of the box, to explain their answer or told that they are wrong they are
thrown off. This was shown through Maddux’s thinking in the third task. He thought he
immediately knew the answer and was happy with that but missed the understanding while
trying to find an “explicit formula” (p 275) for the pattern. Another instance where is
thinking was thrown off was when he was asked to extend the second manipulative in task
one. He could not consider that the pattern could go on to 0 and beyond, insisting that it
“wasn’t possible”. I found this surprising because surely he knows that negative numbers
exist. Maybe the manipulatives threw him off, but we always could have used another color
to represent negative numbers. Maddux’s major understanding shown in the interview was
his recognition that task two’s pattern could be solved with an explicit formula using
multiplication. In his class, they had only dealt with addition and subtraction numerical
patterns. When I made these questions, it had never even occurred to me that one of the
students would make this connection. This showed his higher level understanding of
numbers and numerical relationships. When Kaylee, the 5h grade student who has made
more exposure to multiplication, was asked to do this problem another way she did not
make this connection.
While Kaylee did not have the quick recall and fact knowledge that Maddux did, she
displayed the best understanding of patterns as a concept. She pointed out multiple times
that patterns continued in both directions and used this to her advantage in many of the
problems where Hallee and Maddux struggled. She also could recognize the different types
of patterns by name as we did them (ie saying “this is a growing pattern” on task three).
Another advantage Kaylee had over the 4th graders was that she was more easily able to
Diagnostic Interview
create different methods of computation. In her class they must have more experience
with this because when I asked her for non-standard algorithms it seemed like a normal
question for her. When Hallee and Maddux were asked they acted as if it was a very
strange thing to ask them especially if they had already correctly answered the question and
had to use a lot more time thinking about it than Kaylee.
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Use supporting evidence from the students’ quotations, behaviors, and work during the interview, as well as
your interpretations and notes from the previous section.
Include at least one reference (with page numbers) to a relevant source on math development (textbook,
research, etc.).
Recommend next steps (x/40)
This section should consist of individual and group recommendations, organized as best fits your experience.
 Make specific recommendations for next steps in the students’ mathematical learning.
 Use supporting evidence from the students’ quotations, behaviors, and work during the interview, as well as
your interpretations and notes from the previous section.
 Include at least one reference (with page numbers) to a relevant source on math development (textbook,
research, etc.).
 CITE TEXTBOOK CH 14
 As a group, I think it is impossible to place all of these students in the same boat because
of their differing grades and levels of mathematical understanding. In 4 th grade in their
math class, I think Maddux’s and Hallee’s teacher should revisit the understanding of what
a pattern is. Both students missed the understanding that patterns are continuous and that
they can be extended in both directions. They were unable to add on to the beginning of
the pattern correctly and further, could not see that it was obviously incorrect to follow
the same rule because their patterns visually looked wrong. I think this stems from their
teacher’s math lessons which in my opion only teach surface level math. They are taught
just enough to get by on the SOL test, but not the reasoning behind methods or alternative
solutions.
 On that note the second recommendation I would make for that group would be, not only
in patterns but in all math, would be for them to be pushed to create their own methods of
computation rather than just doing exactly what was shown to them, both of them did this
in at least one problem using “carrots” in between numbers to show the pattern’s rule. If
Hallee had thought of her own method, maybe it would have included checking more than
just two numbers before deciding a pattern’s rule and lead her to be more successful.
 For Hallee I think that pattern understandings need to be revisited. What makes a pattern?
What kind of patterns are there? Can a pattern rule always the same? Can it include
multiple steps? Number sense was a weakness for her which should be continually worked
on. Her addition and subtraction computation has not moved into quick recall and still
relies heavily on manipulative and her fingers. I think this should be reviewed and she
should be given more practice time with this maybe in a pull out or after school program.
Because math understanding build on each other, the more the class moves on as she
struggles to compute these problems the further she will fall behind. I think she has the
capacity for further understanding with patterns but without moving on in the phases of
understanding of these computations she will continue to confuse herself with
computation mistakes.
 Kaylee is alike to Hallee in that she also needs work on her addition and subtraction
computation to move into automatic recall and quick recognition but she has a stronger
number sense and a more developed understanding of patterns. Out of all these students if
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given a test of higher-level thinking questions on numerical problems she would be the most
successful, not because she is the “best at math” or the “smartest” but because of her
broad understanding of patterns and her ability to use alternative methods and nonstandard algorithms. Because of these key concepts, I feel she could work out almost any
problem given to her if she set her mind to it.
Maddux is the opposite to Kaylee; his automatic recall and mental math abilities are
astounding but his higher-level thinking when it comes to understanding patterns- shown by
not being able to extend the pattern in the opposite direction like Hallee- and thought
process of only doing math in the standard way is where he needs work. I belive that if
guided through inquiery, this student could make leaps and bounds in his higher-level
thinking if he was pushed and asked the right questions. He has a very bright mind but a
lack of conceptual understanding and a narrowed focus.