Mrs. Jones put her students into groups of 5. Each group ...macyfraylick.weebly.com/uploads/4/3/5/3/43530345/... · of boys in a group instead of boys in a class for his final answer

IMPACT ON STUDENT LEARNING MACY FRAYLICK 1

INTRODUCTION TO THE TASK

Statement of the Problem

I posed my problem to a group of 10 students in 6th

grade at a field placement. The

problem is as follows:

Mrs. Jones put her students into groups of 5. Each group had 3 girls. If she has 25

students, how many girls and how many boys does she have in her class?

This was done during homeroom in an ELA classroom. The students were exceptionally

fascinated with the idea of doing math in a non-math class and competed over who my 10

subjects would be. They were eating breakfast and watching their morning video of Myth

Busters while doing the problem. They had about 15 minutes to complete it; each student did so

independently with no guidance from me or their peers. When finished, they raised their hands

and I collected their work.

My field experience is in a suburban-urban area in southwest Ohio. Every student here

qualifies for free lunch. The 10 individuals I chose were diverse in terms of gender, ability, and

ethnicity. One student was on the autism-spectrum. As a differentiation suggested by his

teacher, I read the problem aloud to him.

While I was in an ELA classroom, I had the chance to discuss this problem with the math

teacher. She said that, while students have had practice with equivalent fractions, they have not

yet learned proportions. She guessed that about half of the students would get the problem right,

those who conceptually understand math. Looking over the problem, the mathematical

terminology that the students must know and understand is “grouping;” the math teacher said

that this is not a new word or concept for these 6th

graders.


Rationale for Task

My objective was that, through doing this activity, students would be able to solve and

conceptually understand a part-part-whole proportion problem. This is the most simple of all

proportion types; students can solve through informal methods of reasoning. The students at this

school will be learning proportions within the school year; in order to supplement the future

instruction, I posed this problem because it enables students to think conceptually about the

proportions. It places it in a real-life context. This aligns with the Common Core State Standard

which reads, “Use ratio and rate reasoning to solve real-world and mathematical problems”

(6.RP.A.3). My posed task can be solved in a number of ways, from informal methods to formal

proportional thinking. Given the students use of equivalent fractions and general modeling

experience in math, this is a realistic problem to pose. It connects to personal interest because

grouping in class is something that all students experience at school; it can be easily visualized.

This question also relates to all students’ cultural backgrounds as they have exposure to similar

classroom situations in their daily lives.

ANTICIPATED STUDENT OUTCOMES

Langrall and Swafford (2000) discuss 4 different levels of proportionate reasoning.

Level 0 is a method that is random and revolved around guessing; it does not actually

demonstrate proportional reasoning (Langrall & Swafford, 2000). I predict that 2 out of the 10

students will use this method after discussing the prompt with the math teacher. They will most

likely read the problem, be confused by conceptuality of the prompt, and arbitrarily add, subtract,

multiply, or divide the numbers given. None of these answers will be correct. I anticipate that

the work will be similar of that below; I simply multiplied the number given of 5, 3, and 25.

http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf


Even though the question asked for the number of boys and girls, this solution only equals one

value of 375. Assuming that the students who demonstrate this level of reasoning do not

understand what the problem is asking for or choose to ignore it, I simply circled the answer that

was given. My predicted work is shown below:

As stated by Langrall and Swafford (2000), when solving a part-part-whole problem,

students are more likely to use informal procedures, like drawing, even when the student knows

how to solve using higher-level thinking skills. This is a Level 1 proportional reasoning strategy

(Langrall & Swafford, 2000). The numbers in my problem are small and reasonable to draw.

Therefore, the cooperating math teacher and I both agreed that the majority of students would

use this strategy to solve. In fact, Landgrall and Swafford (2000) used this problem as an

example for how students would use informal reasoning strategies to solve part-part-whole

problems. Their example can be seen below:


(Langrall & Swafford, 2000)

I predict that the majority, about 6 of the 10 students, will use the Level 1 approach of

proportionate reasoning. I predict that all of such students will have the answer correct unless

they made a simple mechanical error in multiplication, addition, etc.

Level 2 exhibits reasoning that only students who use and prefer higher-order thinking

will demonstrate (Langrall & Swafford, 2000). Their math teacher and I agree that only a couple

students would use this method. I predict only 2. I think that 1 of these 2 students will use a

mixture of unit rates and multiplication to solve. This student will most likely be the more

advanced individual who fully understands the problem and can demonstrate using quantitative

reasoning as seen below:


Another method that a student might use that is at the same level is quantitative reasoning

through creating a table (Langrall & Swafford, 2000). This is a familiar mathematical strategy

for these 6th

grade students, so I predict that at least one individual will use it to solve:

Number of Girls Number of Boys Number of Groups Total Number of

Students

3 2 1 5

6 4 2 10

9 6 3 15

12 8 4 20

15 10 5 25

I believe that all students that solve using level 2 reasoning will solve correctly because this

requires upper-level thinking that only the most advanced learners will attempt. If one of these

individuals answers incorrectly, it is most likely due to a mechanical error that does not reflect

the student’s actual understanding of the problem and process.


Level 3 refers to formal proportional reasoning in which the student can actually set up

and solve a proportion. The class had not yet learned this, so I predict that no student will solve

using this strategy:

DISCUSSION OF COLLECTED DATA/ ASSESSMENT

Analysis of Student Learning

Below is a summary of student learning.

Student

Number

Level Strategy Correct? Work Notes

1 0 Unable to

link/

random

No

This student began thinking

about grouping correctly, but

did not know what to do with

his knowledge of 5 groups of 5

students. He arbitrarily then

began subtracting the other

values given to him in the

problem from 25. Lastly, He

subtracted the 22 and 20 from

25, incorrectly claiming that it

equalled 3. I believe that this

was a way for him to double

check his work. He never

labeled the amount of boys and

girls, so it seems as though he

is unclear as to what exactly the

problem is asking.


2 0 Random No

This student is on an IEP for being

on the Austism-specturm. It seems

as though he began arbitrarily

adding the numbers given in the

problem. He wrote a number line

to help him add. While he added

correctly, he did not make meaning

of what he was adding or the sum.

3 0 Unable to

link/

random

No

This student was unclear of the

relationship between the 25 total

students and 3 girls in each group

of 5. She divided the 25 by three,

which would tell her how many

groups of 3 there are in all of the

class. This is incorrect because

there are girls and boys in the class,

and they are seperated into 5’s, not

3’s. She knew that there had to be

at least one boy, so labeled her

remainder as being such. While

her computations were correct, her

conceptual knowledge of the

problem is lacking. 4 0 Unable to

Link

No

This student was right in her

method of solving how many girls

and boys are in one group, but that

is not what the problem is asking.

She begins correctly by finding

that there are 5 groups of 5, but she

doesn’t know what to do with that

information or how the 3 girls and

2 boys in each group relates.

5 1 Model Yes

This student understood the

relationship between groups and

the number of boys and girls in

each, she just didn’t label her final

answer.


6 1 Model Yes




each, she just didn’t label her final

answer (interestingly the same as

#5 even though they weren’t sitting

nearby).

7 1 Model No




each, he just looked at the number

of boys in a group instead of boys

in a class for his final answer. He

figured the number of girls

correctly.

8 2 Unit Rate Yes

This student used unit rate of the

number of girls per group to solve

the answer. She is very flexible in

her mathematical thinking about

the problem. She could have

labeled her work more clearly for

the reader.

9 2 Unit Rate Yes

This student used unit rate of the

number of girls per group to solve

the answer. She is very flexible in

her mathematical thinking about

the problem. She could have

bettered her answer by showing

more work to express her thinking.


10 2 Table No

This student understands the idea

of grouping and how many

girls/boys are in each group. She

simply doesn’t know how many

groups were in the class or when to

stop counting. She didn’t know

how to double-check her answer.

This problem requires primarily conceptual knowledge to solve. Procedural knowledge

is used in many ways across student work. In all cases, procedures were done correctly. This

involved simple addition, subtraction, multiplication, and division problems. The error was what

they put into their equations and the meaning that was made from the results. This was largely

due to conceptual knowledge of the problem.

Language Uses

It appears as though the idea of “grouping” was one of the major causes of student error.

While none of them mentioned it in their work, the use of the term was evident. Many students

understood the conceptual idea of grouping (therefore many divided 25 by 5 to figure the total

amount of groups). However, students didn’t know what to do with the result. This can be seen

with student #1. He figured that there were 5 groups of 5, but did not know how to connect that

with the number of girls/boys in each group, so he arbitrarily started subtracting numbers.

Student #3 understood the relationship with grouping and the number of boys and girls in each;

she simply could not proportionally relate it to the whole class. Student #10 used more


advanced thinking and organized the data with a table. While she knew that 3 and 2 made up

one group, she was unsure of when to stop adding these groups. In students 5-6 & 8-9, however,

their knowledge of grouping appeared to be an advantage. Whether they created a model or used

a unit rate, knowing that the students were grouped provided more of a direction for students and

allowed them to quickly solve.

The problem itself did not refer to specific syntax since it was more conceptually-based.

No student was at Level 4, so the students only used the basic syntax of addition, subtraction,

multiplication, and division. Therefore, it was not a matter of students being able to do these

computations that caused errors, but the ability to set up the equations and to make meaning of

the results. An example of syntax results being used incorrectly is with student #1 who correctly

computed the different equations, but had trouble making meaning of the results. This contrasts

with student #8 who was able to set up the equations, used syntax correctly, and made meaning

of the results; this led to the right answer.

Feedback

When writing my feedback, I made comments on their strengths, as well as non-leading

comments on parts of their work that, when re-evaluated, should lead the students to the right

thinking about proportions. It should clear up conceptual knowledge and lead to correct use of

procedural knowledge in order to solve. I will provide an example of my feedback for 3

different students of the 3 levels that were used:


Level 0

This student (#4) was unable to link the relationship between the amount of boys and

girls per group and the total amount of students in the whole class. I first highlighted her

strengths by saying that I liked the way that her work was coming. She was able to find the

number of boys and girls in each group, and that is a good step. In order to get her thinking

proportionally, I asked what the 5 represented (she is so close to moving onward with her

thinking!). I also put a comment about the whole class to move her critical thinking forward.

She should be able to use the feedback to deepen her conceptual and procedural understanding.


Level 1

I really liked this students’ (#7) method of labeling the amount of boys and girls, and

therefore highlighted this strength. For his answer, he thought of the girls in terms of the whole

class, but the boys as each group. To move his learning forward in terms of proportions, I asked

him to explain the meaning of the whole numbers. In addition, I asked him about finding the

final values as this should cause him to re-analyze his work in order to figure that he looked at

the number of boys vs. girls from two different perspectives. This comment will push him to

look back at the relationship between his work and his final answer. This will deepen conceptual

knowledge and will push him to use procedural knowledge in a correct way that will guide him

to the right answer.


Level 2

This student (#10) used a table to solve, which I thought was a great strength. It showed

me that she was on the right path to organizing her thoughts in order to solve; she just was

lacking some of the conceptual knowledge in order to know when to stop with the rows.

Therefore, I asked what each row represented and how she knew when to stop. This should

make her realize that her method shouldn’t be arbitrary, but one with a clear and succinct

purpose. I also asked her what the 24 and 14 represent because they clearly add up to more than

25 students; she never even labeled her answer. The comments should make her re-evaluate her

thinking in such as way that leads her to correct her conceptual misinterpretations. This should

also cause correct procedures that produce the right answers.

REFLECTION/ IMPLICATIONS REGARDING COLLECTED WORK

Comparison of Prediction and Collected Data

Overall, my predictions were very close to what my students produced. I predicted that 2

students would be at Level 0 who arbitrarily add, subtract, multiply, and divide the numbers

provided to solve. Surprisingly, 4 actually were at this level. One of such randomly computed


as I had predicted, while the others had a mixture of randomly computing and not being able to

make connections between the group and class proportion. I predicted that 6 students would use

a model to solve, but surprisingly, only 3 students did. I also thought that, of students that used

the model, their answers would all be correct. Unexpectedly, 1 of the 3 drew incorrect

conclusions about their work and did not arrive at the answer. For Level 2, I predicted that 2

students would solve this way, one using a table and the other using units. I was close, except

for 3 total were at Level 2, 2 of which that used rates and 1 that used a table. Again, I thought

that these students, because they had more advanced conceptual knowledge, would get the

correct solution. Surprisingly, the student who made a table drew incorrect conclusions about

the results and arrived at the wrong solution; it was more than a simple computational error. I

was right when figuring that no student would use Level 3 as they have not yet been formally

taught proportions. Overall, I would say that I was most surprised that so many students were

either at Level 0 or at Level 2. Although Level 1 is informal and accessible for all students, few

chose this route. This could also be because the math teacher does not use models very often to

supplement instruction.

Next Instructional Steps

Although students are scattered across different levels of the proportional reasoning, I

would build off of this task by having students in my class actually stand up and model this

problem. This would help conceptualize the problem and would show the relationship between

the groups and the class. Students could share their many methods to solving. This would also

provide a great transition into teaching formal proportional reasoning (Langrall & Swafford,

2000). Having students, from here, develop a formula would be a great way to develop

procedural knowledge about proportions. This would also build problem solving skills as they


are making meaning of the relationships in proportions themselves. I might also present

associated sets or well-known sets to students as they formulate their proportion computations

(Langrall & Swafford, 2000). This would be a meaningful way to move from informal to formal

proportional reasoning.


References

Langrall, W. C., & Swafford, J. (2000). Three balloons for two dollars. Mathematics Teaching

in the Middle School, 6(4), 254-261.

Mathematics standards. (2015). Retrieved November 6, 2015, from

http://www.corestandards.org/Math/

http://www.corestandards.org/Math/

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Mrs. Jones put her students into groups of 5. Each group ...macyfraylick.weebly.com/uploads/4/3/5/3/43530345/... · of boys in a group instead of boys in a class for his final answer