CGI Math Interviews

Meghan L. Williams

Kennesaw State University

INTRODUCTION

I completed this assignment with Luke, a first grade student at my school. Luke is a

Caucasian student who has been in a dual immersion program for three years, so he has received

math instruction in English and Spanish, though English is his first language and the only

language spoken at home. He is an average student overall, but math is his strongest and favorite

subject. He was a little weary about completing the activity with me at first, but warmed up to

the experience after the first problem.

PERFORMANCE ON PROBLEMS

Luke was successful at solving four of the six problems presented. He used the counting

block manipulatives in the last five problems. With the first problem, Luke completed it

correctly using mental math by using a derived near double strategy, though he could not explain

how he knew to use the double strategy (Carpenter et al, 2014). He was especially successful

with problems having a result or whole unknown and one problem with a change unknown.

Luke struggled with the two problem requiring him to determine “how many more.”

Though one of the problems was a join problem, comparing the current amount to the needed

amount was something he could not successfully understand. However, he did model both in

ways that could have been successful. With both problems he made piles of blocks representing

both amounts. With the first problem, he transferred the number needed to make the twelve, but

instead of counting how many blocks he transferred, he stated the number he ended with. In the

comparison problem, he tried to take away the lower quantity from the higher quantity, but

miscounted and thus ended up with an incorrect answer (Carpenter et al, 2014).

UNDERSTANDING OF ADDITION AND SUBTRACTION & NUMBER SENSE

I definitely feel like Luke has a solid understanding of addition and subtraction. Though

we did not discuss those particular words, he did say things such as “put them together” and

“take them away.” In all the problems, he did seem to understand the necessary actions to solve

the problem correctly. His errors really came in with careless counting errors that he made in

two of the problems. I do not know exactly where his class is with their instruction of addition

and subtraction, but I feel confident that Luke has a solid understanding.

In the first problem, I felt Luke had a good understanding of number sense. When he

explained how he arrived at his answer, he explained that 5 + 5 = 10, and when you take 1 from

10 you get 9. However, when I asked him why he took away 1 or did 5 + 5 he could not explain

his thinking. This is the only problem I noticed him working with derived facts, so I definitely

think his number sense is developing.

NEXT STEPS & COMMUNICATION TO PARENTS

I would certainly continue working with Luke on comparison problems as well as

problems that have the change unknown. I do know that our first graders have just started

comparison, so it was no surprise that he struggled some with that particular question. His

understanding of the action involved in the story problems seem solid. I would work more with

him on counting or modeling carefully and encourage more use of derived facts as he feels

comfortable.

Since Luke’s mom is a teacher at the school, I did actually have the opportunity to share

the results of this activity with her. I shared Luke’s strengths in being able to model the

necessary actions in the story problems and his use of the near double strategy. I encouraged her

that he needs to continue practice with the comparison problems and being very deliberate and

careful in his problem solving (Aguirre et al, 2013).

References

Aguirre, J. M., Mayfield-Ingram, K., & Martin, D. B. (2013). The impact of identity in K-

8 mathematics learning and teaching: rethinking equity-based practices. Reston, VA:

The National Council of Teachers of Mathematics, Inc.

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2014).

Children’s mathematics: Cognitively guided instruction. (2nd Edition). Portsmouth, NH:

Heinemann.

I worked with Angel, a second grade student who is in a bilingual immersion class.

Angel is new to the school this year having been previously in Texas. He expressed that he was

good at and liked math and was very interested in helping with the project and asking if he was

the only one selected. Of the six problems, he correctly answered four.

STRENGTHS & WEAKNESSES

Angel answered all three division questions correctly, using either direct modeling or

derived facts. The partitive division problem seemed to be one of the easiest problems for him to

answer as he did so quickly and correctly using derived facts. He used direct modeling for both

measurement division problems, but easily and correctly answered these questions. He even

understood the meaning of the remainder for the final question.

The multiplication problems surprisingly presented the most difficulty for Angel. He

only answered one of the three correctly and did so by using a fact he had already memorized (4

x 10). He attempted to use counting strategies for the other two multiplication problems and

made errors both times in his use of manipulatives. For the first problem, he set out one more

block than necessary and ended up counting it with his skip counting. For the fourth problem, he

seemed to start using derived facts then mixing up those facts with the original numbers and thus

making a mistake in how many of each group he should count.

UNDERSTANDING AND NUMBER SENSE

Even with his errors, I felt that Angel did understand the process of multiplication and

division. With every problem, he was able to at least begin to model or explain it correctly. He

certainly seemed to understand the action of each word problem and was moving in the direction

it required. He used all three types of strategies, so he is clearly in a stage of still figuring out

multiplication and division and work with them efficiently.

Angel also seemed to have strong number sense. He used skip counting by fives and

knew at least some facts with multiplying and dividing by ten by doing 4 x 10 and knowing 30

divided by 3 was 10. He was easily able to manipulate the numbers and understood remainders

and possibly when he had an incorrect remainder. In question 3, he had accidently counted out

37 blocks instead of 36, so when he ended with one remaining of the groups, he recounted the

whole and realized his earlier mistake and quickly put away the extra block.

NEXT STEPS & PARENT COMMUNICATION

With Angel, I would consider more work with multiplication. A problem I might give

him is: Julia made cookies for her 5 friends. She gave each friend 4 cookies. How many cookies

did she make? With this multiplication problem, he could easily model it with counters or even

use his skip counting strategy to determine the answer. By keeping the numbers smaller and

within his range of numbers he is successful with, he can focus more on deciding which strategy

would be effective. As he is successful with the strategies, he could begin working with larger

numbers.

Angel presents an interesting case when communicating with parents because he uses so

many different strategies for his problem solving, each of them valid. I think when talking to his

parents, I would communicate with them a few of the more effective strategies that they may

encourage him to use at home. As it stated in the chapter, the important aspect is that he is able to

communicate his thinking, which he seemed able to do in our conversation. I would encourage

his parents to give him problems, especially multiplication problems, as they arise around the

house and have him explain how we solved the problem.

For this assignment, I chose John, a low performing third grade student. John is in an all-

English classroom and receives ESOL service. He is underperforming in all academic areas.

Though he was provided counting block manipulatives, he chose to work all his problems on a

dry erase board.

STRENGTHS & WEAKNESSES

John correctly answered 4 of the 6 problems presented. In all but the first problem, he

chose to do a strategy that involved addition. This definitely seems to be an operation he feels

comfortable with. He was successful with regrouping in each addition problem. For the

multiplication problem, he chose to add 10 five times rather than recalling a multiplication fact

or even seeming to think of it as multiplication. Even for the measurement division problem, he

chose a very inventive strategy of adding tens and threes until he got close to 100.

John struggled with the first problem which required subtraction with regrouping. He

paused when he realized the problem was going to require him to do 8-9, and instead did 9-8 to

get 1. Regrouping did not seem to come to his mind as an option. He also struggled some with

the last problem. Though his strategy was inventive and had the potential to lead him to a

correct answer, he did not ensure to keep the same number of tens and threes. He also came up

with the total of 102, so he just subtracted 2 to get his desired 100.

UNDERSTANDING OF NUMBERS

When it comes to adding multi-digit numbers and regrouping, John did not seem to have

a problem. He also seemed to understand required operations to solve the problems. His

number sense was difficult to get a grasp on. As stated, he struggled with the subtraction with

regrouping. At first, I was skeptical of how well he was going to be able to perform with the

more challenging multiplication and division problems since he was not able to correctly answer

the separate problem. As we went on, however, I was pleasantly surprised at his inventive

strategies. For the compare problem, his thinking was flexible enough to see that if he started

with the younger age and counting on to the older age, he could tell the difference. This allowed

him to add, which is an operation he felt more comfortable with. He also saw the possibility for

adding in the division problem to add up to what his target was. His number sense did show

weakness, though, when he was unable to understand the issue in not getting exactly 100.

NEXT STEPS & PARENT COMMUNICATION

I would definitely encourage more work with division problems, though I might start

with a partitive division problem since those tend to be simpler for students to understand in

knowing the number of groups (Carpenter et al, 2014). A sample problem for John may be:

Jenna bakes 96 cookies. She wants to give each of her friends 12 cookies. How many friends

can she give cookies to? This would allow John to continue his addition strategy in a problem

where he would get the target number and simply have to count how many groups of 12 he used.

This could strengthen his understanding of his own strategy.

In a parent conference, I would certainly communicate to his parents his struggles with

subtraction with regrouping and encourage this to work with this at home. I would also share his

comfort with addition and how he can use that strategy with many problems. By communicating

with his parents his inventive strategies and encouraging them to have him explain his thinking

at home, John can have opportunities to strengthen his strategies so that he is able to understand

when and why these strategies are effective. By inviting parents into the conversation about

mathematical thinking beyond typical operations associated with word problems, John’s

mathematical thinking can be encouraged and grow at school and at home (Aguirre, Mayfield-

Ingram, & Martin, 2013).

References

Aguirre, J.M., Mayfield-Ingram, & Martin, D.B. (2013) The Impact of Identity in K-8

Mathematics: Rethinking Equity-based Practices. Reston, VA: National Council of

Teachers of Mathematics.

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2014).

Children’s mathematics: Cognitively guided instruction. (2nd Edition). Portsmouth, NH:

Heinemann.

For this assignment, I worked with Edward, a fourth grade student. Edward is Hispanic

and currently receives ESOL services. He is a high-average math student typically, but he is

sometimes inconsistent in performance based on the standard.

STRENGTHS & WEAKNESSES

Edward was able to solve every problem pretty successfully. In the first problem, he

originally made a mistake by saying “5.” When I asked him to explain his thinking, he worked

through the counting strategy of counting by halves and realized he made a mistake by only

counting to ten cupcakes, not twelve. He then corrected to “6.” He was also very successful in

visualizing and rarely used the given worksheets. He also understood the efficiency of giving

out as many wholes as possible before separating into fractional pieces. He shared that with

problem 2, he wanted to give a whole brownie to each person but realized that would only be 4

(people) so he had to split them. With the cookies in problem 3, he first “gave everyone one”

and realized there would be 3 left to be divided.

The only incorrect solution Edward had was on the last problem in which he had to

divide into sevenths. He first divided into even eighths and realized he had one piece too many.

He then erased one line but saw that it created one big piece. Instead of readjusting all pieces, he

only adjusted the lines of the pieces on that half of the circle. He stated that the pieces looked

the same size, so he does have some trouble understanding equivalence in creating equal sized

pieces.

UNDERSTANDING OF FRACTIONS AND NUMBERS

When answering, Edward rarely used the fraction terminology. He would refer to halves,

but when explaining his drawings he would state “3 pieces” or “4 pieces” instead of thirds and

fourths. I inferred from this use of language that Edward does not feel completely comfortable

with fractions at this time. I also noticed this with his drawing of sevenths. He seemed to realize

the importance of the pieces “looking the same,” but his rational beyond that was lacking.

I do feel Edward has good number sense. He was flexible with the numbers and saw the

relationships between the 4 brownies and 8 students and the 3 leftover cookies among 6 students.

He quickly saw that dividing the objects in half would give him the desired amount. While he

did use a counting strategy for the first problem, he did understand the need to count by halves

and could do so successfully.

NEXT STEPS & COMMUNICATION

Since Edward was so successful with the given problems, I would probably supply him

with an equal sharing problem where he would have to separate the objects into partitions other

than halves. As Empson and Levi (2011) suggest, students often see fractional parts as halves.

Their understanding of fractions sometimes falls short beyond that. A sample problem I might

provide is: Five brownies are being shared among eight students. How much of a brownie will

each student get? Since Edward is an ELL, I would keep the wording simple, but I do provide a

challenge in providing the numbers in written form (Empson & Levi, 2011).

In communicating with Edward’s parents, I would explain his strengths in being flexible

with his thinking. I would encourage them to allow him to work things out with inventive

strategies and encourage them to allow him to explain his thinking as he may be able to find any

mistakes he made by doing that. I would also encourage them to work with him on partitioning

objects into unfamiliar denominators, such as fifths or sevenths, or partitioning irregular objects.

This can allow him to practice justifying equal partitioning beyond the memorization of

partitioning fourths by drawing a horizontal and vertical line down the middle (Aguirre,

Mayfield-Ingram, & Martin, 2013) .

References

Aguirre, J.M., Mayfield-Ingram, & Martin, D.B. (2013) The Impact of Identity in K-8

Mathematics: Rethinking Equity-based Practices. Reston, VA: National Council of

Teachers of Mathematics.

Empson, S. B. & Levi, L. (2011). Extending Children’s Mathematics: Fractions and

Decimals. Portsmouth, NH: Heinemann.

I worked with Victor to complete this assignment. Victor is an average student in math

and reading. He does not qualify for any extra academic services, but he is fluent in Spanish and

English. He typically takes pride in his academic work and sees himself as a good student.

STRENGTHS & WEAKNESSES

Victor showed a 50% accuracy rate with his answers, though he did show reasonable

problem solving strategies with each problem. When given problems with fractions in the

problem, he clearly used his understanding of fractions to complete the problem. When the

problem supplied only whole numbers, he did not resort to fractions to solve, even when it was

necessary. Victor quickly saw the patterns in the first and last problems. He created a sort of

table for both problems the label each notebook/student to count up the needed money/paint.

With the last problem, he saw the pattern within four students, though added a couple middle

numbers before coming up with the final answer of 25 thirds. He also was strong in his visual

representations of fractions. He used drawings to compare and ⅓ and referred to ⅖

visualizations of fractional pieces to order the fraction cards.

Victor seemed to focus solely on numbers and very little on what the numbers

represented. With the question asking how many loaves of bread he could make, he saw that 6

doubled would be 12 and automatically doubled 8 to get 16. However, he did not take into

account that the 6 represented loaves of bread and the 12 referred to cups of flour. With the

running question, he understood that for every mile 10 ounces of water were consumed, but he

simply stated that 4 ounces would remain without considering any more could be run.

UNDERSTANDING OF FRACTIONS, DECIMALS, AND NUMBERS

In this interview, Victor never showed any use of decimals so I could not judge his

understanding of them. Even with the first question of the use of money, the answer and steps

between could be solved with only the use of whole numbers which is what he did.

With fractions, Victor seemed to rely heavily on visual representations of fractions. In

comparing and ⅓, he drew a model of both to see that was slightly bigger. Though he was ⅖ ⅖

correct, he did not take into account his drawing could be inaccurate. In order fractions, he

ordered them from largest denominator to smallest saying, “if you draw them, 12 is the smallest

piece, 8 is the next smallest… then 4, and 2, but 3/3 is one whole.” The only time he seemed to

take into account the numerators is when he had two fractions with eighths and when he saw 3/3

was one whole.

Overall, I would certainly say his number sense is developing. Victor seemed

comfortable in finding patterns to solve the problems but did not show much flexibility in his

thinking. With each problem, he would silently thinking about it for a moment and write down a

strategy which he would continue until coming to a solution.

NEXT STEPS & COMMUNICATION

As stated above, none of the questions required Victor to show an understanding of

decimals. To test this, I might give a problem such as: Jennifer wants to buy 5 donuts that cost

$1.35 each. How much money will the donuts cost? This problem will require an answer that

uses decimals and allows for Victor to use either multiplication or repeated addition to respond

and will require him to show understanding of decimals following base ten principles (Empson

& Levi, 2011).

For our next steps, I would communicate to parents Victor’s strengths in recognizing

patterns and visualizing and needing to extend his reasoning of seeing whole numbers able to be

used as fractions, particularly with the use of remainders in division problems. I would hold

Victor responsible for always referring to objects of numbers (6 loaves, 12 cups, etc.). I would

encourage his parents to also encourage him to label numbers with objects words and provide

problems that would require him to divide whole objects (e.g. sharing five cookies between him

and his brother). I would resolve to working with him on different ways to interpret remainders

(as fractions, decimals, or representing another group, etc.) (Aguire et al, 2013).

References

Aguirre, J.M., Mayfield-Ingram, & Martin, D.B. (2013) The Impact of Identity in K-8

Mathematics: Rethinking Equity-based Practices. Reston, VA: National Council of

Teachers of Mathematics.

Empson, S. B. & Levi, L. (2011). Extending Children’s Mathematics: Fractions and

Decimals. Portsmouth, NH: Heinemann.