Arica Fowler
March 31, 2017
Diagnostic Interview
Part 1
Kohen is a second grader in a general education classroom. He is quiet and very well
behaved, always on task, and usually gets his work done quickly. He is in the highest reading
group. He is on grade level in math. He seems to have a good number sense. Typically, during
math warm up activities, he solves math problems mentally, and I have noticed him using the
hundreds chart to solve. His answers are usually correct.
Rachel is also in a second-grade general education classroom. She is also quiet and well
behaved. She seems to struggle in all subject areas however. She will not complete her
assignments when others have and usually must work with the teacher to get them done, not due
to a lack of effort but due to a lack of understanding. She is defiantly not confident in math and
below grade level. She rarely offers answers to the math warm up questions. During a
subtraction activity, I noticed she does not understand re-grouping. I noticed she had written a
one above all the tens places and when I asked her to explain that to me, she said “you are just
supposed to” and couldn’t explain the reasoning. She gets pulled for math remediation as well.
Part 2
2.7 The student, given two whole numbers, each of which is 99 or less, will
a) estimate the difference; and
b) find the difference, using various methods of calculation.
The focus of this interview is to determine the mathematical understandings of these two
students, in regards to place value using subtraction. I will pay close attention to how they solve
the problems and ask specific questions to determine their understanding. This will let me
know if they are at a pre-place value state or if they have a unitary, base ten or equivalent
understanding of place value. I will use subtraction problems to help me do so by seeing if the
students count by ones to determine the difference or if choose to group numbers by their
value to solve the problems. If they are on the verge of mastering one of these stages, I will
then determine the next steps to take to move them along in their mathematical
understanding.
Student Task
-Tell the student to show you 15 using
manipulatives. (base 10 or counters)
-Write down 15-3. Have the student solve
using the manipulatives.
-Have them write down the answer.
How do you know that is the answer?
Can you solve this problem another way?
-Ask the student to solve this problem
using a drawing/picture, manipulatives.
-Derrick had 27 pokemon cards. He gave
9 of them to his little brother. How many
does he have left?
Can you explain your drawing?
Can you solve this another way?
-Give the student this problem and let
them solve it how they wish.
-There were 70 students in the second
grade at Sunny Elementary School. On
Monday, only 65 students came to
school. How many students stayed
home?
How did you find that answer?
Can you find it another way?
How do you know it is correct?
Interview Notes
-Write down 64-38. Tell the student to
solve the problem. (Try to get them to
solve symbolically)
How did you get that answer?
How do you know it is right?
Can you solve this another way?
(pictorial or manipulatives)
-Tell the student they can solve this
however they would like.
-Amber made 81 cookies for her family
cookout. 55 of them were eaten. How
many does she have now?
Why did you choose to solve it that way?
Can you explain what you were thinking?
Can you solve it another way?
(pictorial, concrete, symbolically)
-Tell student this one is a little tricky but
you think they can do it.
-Write 102- 76.
-They can solve it however they would
like.
How did you get that answer?
*Make sure there is at least one concrete representation, one pictorial representation and one
symbolic representation.
*Make sure there are at least two connected representations.
Part 3
Part 4
I was a little surprised at some of the answers and methods that Kohen gave during the
interview. The last problem he solved does not match up with his typical work, so I am not sure
if he was rushing through or simply just made a mistake that he didn’t catch. However, for the
sake of this assignment I will only analyze the work shown during the interview. His work
overall demonstrated an understanding of benchmark numbers and place value. At first, he
solved problem one mentally and knew that he could take ten away from 27 because 10 was
close to 9. This shows that he is comfortable with his benchmark numbers, in this case the
number ten. He then knew that because he added 1 to 10, he then needed to take that 1 away
from 17. When I asked him to solve the problem using manipulatives, he used base 10 blocks
and counters together, rods for the tens place and counters for the ones place. He knew that the
rods consisted of ten without having to count them. His decision to just cover two cubes in the
rod with his hand and count the rest was interesting. According to Van de Walle, Karp, and BayWilliams (2016) students at the ‘equivalent stage’ are able to “group the pieces flexibly into
versions that include tens and ones but all trades have not been carried out” (p. 225). Therefore, I
believe he is at an “equivalent” base ten grouping stage, because he did not need to break apart
the rod into 10 cubes to take away the two ones. For the second problem, he showed his
understanding of benchmark numbers again, but this time with five. I created this problem to see
if the students would be able to recognize that the two numbers were just five apart or if they
would have to actually solve it to find the answer. Kohen answered this problem almost
immediately. When I asked him to tell me how he knew, he stumbled a little and said, “I don’t
know how to explain it, I just know.” I then asked him to try and tell me what he did in his head.
He stated that he used the hundreds chart, even though he never looked at it when I asked the
problem, and saw that 70 was 5 more than 65 so the answer was 5. This shows he has a strong
relational number sense because he is comfortable with ‘greater than’ and ‘less than’ and using
his benchmark numbers to solve problems. I believe the hundreds chart just helped him explain
his thinking even though he didn’t use it to solve the problem. The problem that seemed to trip
him up was one that forced him to solve symbolically using an algorithm. This was not typically
something that he would struggle with but I wrote the problem on his paper and told him to solve
it using a way they learned in class, but not using manipulatives. When I asked him how he
solved the problem he said that he counted by tens starting at 38. This would have been a good
method but it still would not be solving symbolically and he got the answer wrong. This is where
I think he began to rush. He divided the numbers into tens and ones on his paper so I initially
thought he would solve using the standard algorithm. When he wrote 6 in the ones place I
thought maybe he was borrowing by turning the 4 into 14 to take away the 8 but he never wrote
that down. He then subtracted 3 from 6 and got 3. We ended up running out of time, so I was
unable to ask him more questions. I would have asked him specifically how he got the 6 in the
ones place. If he simply counted up wrong from 38 or if he did borrow, me asking him may have
brought his attention to the mistake. If not, I would then have a better understanding of what
exactly he did. Even though he solved this incorrectly, I do not think this sets him back in his
mathematic understanding because he still seems to grasp the concepts of place value and how to
use that, along with benchmark numbers to solve problems.
Rachel’s understandings were very different from Kohen’s. Her answers to the interview
problems were pretty expected, however based on my prior observations. They showed a very
weak number sense and this therefore puts her at a pre-place value understanding. For the first
problem she demonstrated this unitary understanding because she used the counters to create 15,
then took 3 away and recounted her counters individually. When I asked her how she got her
answer she said that she knew 11+4=15, but these numbers were not a part of her problem. After
she said this, I said “okay, so what is 15-3?” She then pushed all her counters back together and
recounted and said the answer was 12. She then did the same thing when solving with her
drawing. A student with a pre-place value understanding, or a student in the ‘unitary stage’ will
rely heavily on unitary counts and avoid groupings to identify quantities (Van de Walle, Karp, &
Bay-Williams, 2016, p. 224). Her choice to avoid the base ten blocks or grouping of any matter,
made it clear that she has not moved onto a base ten grouping stage. This same understanding
held true for the second problem. She chose to use the hundreds chart first. I saw her count one
by one back from 27. She demonstrated her unitary understanding again when I asked her to
solve using a picture or with another method they learned in class. For the next problem, I
intentionally chose larger numbers so the students wouldn’t try to draw a picture or use counters.
I chose to use benchmark numbers to see if the students recognized the relationship between the
two, or if they would try to solve with an algorithm. Unlike Kohen, Rachel did not recognize the
relationship at all. She said it was a “really hard” problem and she began to shut down. She
would just look at me with an uneasy look and began to say she didn’t know the answer, so I
stopped the interview and told her it was ok and that she had done enough. I believe this problem
showed the most about her mathematical understanding. It is clear that she relies on visually
seeing a quantity, and removing an amount to recount the remaining, in order to solve problems.
When the numbers get too big, she is lost. Both of these components prove her pre-place value
understanding. She drew the line between the tens and the ones places in the third problem, so I
initially thought she was on the right track, but I think she only did this because she saw it done
in class and did not really understand what it was there for. She tried to subtract 5 from 0 and
wrote 5 as the answer for the ones place. This relates to her understanding of number sense,
specifically the relations core. Because she thought it was possible to take 5 from 0, it shows her
misunderstanding of needing something to be able to take something away. Her frustration with
the problem and inability to solve shows that she lacks a strong number sense and this in turn has
affected her place value understanding.
Kohen’s ability to do mental math puts him at a much higher conceptual understanding
than Rachel. He has moved away from needing concrete representations to solve problems and
instead can break numbers apart to help him solve symbolically. Rachel, on the other hand,
requires the concrete manipulatives or some sort of pictorial representation. She has to see the
quantity in front of her to be able to solve. Both of these understandings are rooted from their
number sense. Kohen’s ability to see the relationship between numbers helps him when solving,
while Rachel has not made these connections.
Part 5
In order to move Rachel from this unitary stage to a base ten stage, we would need to
have her begin to look at how grouping things can make solving problems easier. This idea of
grouping will then lead to her understanding of place value and what the ‘tens place’ really
means. Her drawings during her interview show that she does not think of numbers in groups.
When she drew her circles for problem number two, she just drew as many circles on the line
that would fit instead of drawing maybe 10 on the first line, and 10 the second with the rest on
the third. This would have been easier for her to count at the end. I would start off the idea of
having her count in groups by giving her a large group of items. I would guide her thinking by
asking her to think of a way she could count the items instead of counting by ones. I could
suggest counting by a number like 8 because it is hard to count by 8’s. She would hopefully
suggest another number like 2 or 5. I would then say that I think there is another number we can
count by that would be much easier, trying to guide her to choose ten. After she is comfortable
with counting by tens, I would move onto another activity that would have her focus on writing
her number words and connecting those words to a visual representation. To solidify this concept
of groupings, I would have her complete an activity, like the ‘Groups of Ten’ activity from the
textbook (Van de Walle, Karp, & Bay-Williams, 2016, p. 229). In this activity Rachel would
have to count the items in bags and write the number word that represents the amount. She would
then group the items into as many groups of ten as possible and finally notate how many ‘tens’
there were and how many singles. This will help her understand place value because she will see
that the way we write numbers past the teens is connected to place value.
For Kohen, since he seems comfortable with mental subtraction and with using invented
strategies, I would try to get him to move toward solving with symbolic strategies. When solving
the third problem, he tried to solve mentally, but the numbers may have been too large for him
and that might have tripped him up. Therefore, I would start him off with using some of the
strategies he knows, but have him write down his thinking as he goes through the problem. For
example, when solving 64-38, since he said he was ‘counting by tens starting at 38’, I would
possibly show him the number line strategy. This would help him keep track of how many
‘jumps’ he makes. The goal is for him to write down his steps. This will hopefully help when
beginning to learn the standard algorithm, since it consists of keeping track of your work.
Because he is comfortable with manipulatives, in order to teach him the standard algorithm, I
would have him use the base ten blocks to start with, like the example in figure 12.21 from the
textbook. After he is comfortable with using symbols and writing down his thinking, I would
move him away from using the manipulatives to a “complete use of symbols” (Van de Walle,
Karp, & Bay-Williams, 2016, p. 269). I would want him to eventually be comfortable with the
use of an algorithm so that he sees the benefit when moving to different forms of computation,
multiplication and division. I wouldn’t push the standard algorithm because some other methods
may be faster with other numbers, but would want him to be able to solve the problems while
keeping track of all that he does in the problem.
Part 6
This assignment really involved our mathematical knowledge for teaching. We had to
have a strong pedagogical content knowledge to be able to come up with the problems we were
going to ask the students. These problems had to be thoroughly thought out and ordered so that
each question had an element that built on the one prior. We had to think about what we knew
the students’ strengths and weaknesses were, so that the problems were developmentally
appropriate. We also had to plan for the questions we were going to ask so we could dig deeper
into their understandings. Without this knowledge of how the content relates to students and
teaching, we would not have a successful interview. The problems may have been too hard or
may not have had a connection to one another and therefore would not have helped us get an
understanding of the students’ mathematical developmental level.
This assignment aligned specifically with both overarching objectives. We were able to
develop understandings of mathematical ideas children construct because we saw exactly how
they solved problems. We could see how they think about math in general based on the
explanations they give while clarifying their work and how they approach problems. This is
important because no two students are alike. Therefore, we are bound to see very different
understandings even within the same concept. When this happens, we need to be able to identify
what it is they are doing so we know how to facilitate that thinking further or reconstruct their
thinking if there is a misunderstanding. We also refined our own identity and positionality
related to teaching and learning math by seeing how these methods help to understand students.
Using a diagnostic assessment is very beneficial when trying to see where students really need
help. Sometimes it may seem as though they understand the content but when you dig deeper,
you may realize that they have been skimming by due to rote memorization, but have not fully
grasped the concepts. Finally, this assignment also helped us become familiar with and begin to
use formative assessment techniques that inform the teacher’s facilitation of mathematical
learning experiences by having us create our own formative assessment. It takes a lot of practice
and experience for teachers to know what questions to ask and how to ask them. This experience
helped us by allowing us to see our own strengths and weaknesses so that we can better assess in
the future. Some students may need more support when asked more difficult questions, and
knowing this before hand could help an assessment run more smoothly and prevent a student
from shutting down. Overall, the experience was definitely a learning one. I feel as though I have
grown as a future educator and feel more prepared to help my upcoming students.
References
Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2016). Elementary and middle school
mathematics: Teaching developmentally (9th ed.). Boston, MA: Pearson.