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Comprehending Mathematics

Carlye Carson2nd Grade Resident

Pizzo Elementary [email protected]

Comprehending Mathematics 1

mailto:[email protected]

Table of Contents

Literature Review

Teaching Context

Rationale

Question and Sub-Questions

Data Collection

Data Analysis

Claims

Implications

Future Wonderings

References

Appendix


Literature Review

I read through and analyzed several articles, books, and teacher inquires to help

me better understand what data I should collect for my inquiry, how I should analyze my

data and how it should be organized. As I was reading I found strategies that could be

used for helping students comprehend math word problems, ways of collecting my data

and reading skills that improve math skills.

Since I am working in a 2nd grade classroom, some of my students are still

emergent readers. According to Emilie Parkers inquiry, Reading for Emergent Readers,

strategies that work best to improve language/reading development with developing

primary readers. She found out how to decide which strategies to work with, and which

strategies will help them with reading in the next grade. After assessing the students she

was going to work on she worked on chunking and comprehension strategies. She used

KWL charts, background knowledge, and flash cards. Her claims are Graphic organizers

and outlining techniques such as highlighting helps students’ comprehension, silent,

paired and teacher readings help comprehension, and chunking helps children to decode

words, which will then lead to greater comprehension. Pre-reading and scaffolding are

also great for children who struggle comprehending. My understanding of her claims is

that KWL charts, background knowledge, flash cards, graphic organizers, scaffolding,

chunking, pre-reading, and reading are all strategies used to help strengthen student’s

comprehension. The outlining of texts and graphic organizers help with the organization

of a text. Silent reading, teacher reading, and paired reading help children comprehend

informational stories and articles. Overall, comprehension skills will increase a child’s

ability to internalize and apply the new information also increases. This was a good


inquiry for me to read because it gave me great strategies to use with my students to

increase their comprehension of math word problems and not just a text from an article or

story. I also found that it is important for students to figure out what information is

important and what is not important first. Scaffolding Instruction is a strategy that can

also be used by teachers for students so the student can learn to read at their appropriate

level and increase their comprehension (Parker).

After reading “The Day Math and Reading got Hitched” written by Foster and

“Improving Reading to Improve Math” by Glenberg it supported my wondering if

reading and math are connected. According to Glenberg, when children are solving story

problems, errors that they make are not always numerical calculations. Sometimes

students struggle with the language comprehension skill. Children often rather just add

numbers together that they see in a story than make sense of the story. The authors use

the term “sense making” which means to understand the story and what the question is

asking. One of the problems of sense making for children is the inability to pick out

relevant information and irrelevant information. The claims in this article are that

fundamental reading skills should improve reading comprehension across multiple

domains and children face difficulties with picking out relevant and irrelevant

information. They then conducted a study of web-based programs, Uncle Isaac ASIMO

(Moved by Reading), and found that the application of Moved by Reading significantly

reduces the inappropriate inclusion of irrelevant numerical information in the solution

procedures. All in all, they stated and proved that reading and math are connected. The

better a students gets at reading, the easier it is for them to understand math. There is


reading and explaining in math just like in reading. The more you practice reading and

writing the easier it is to read and explain in mathematics (Glenberg).

I found several strategies that could be used in a classroom by teachers to help

students understand what they are reading. Some examples are: repeated readings,

background knowledge, dictionaries, KWL chart, flash cards, eliminating excessive

details, working in pairs/groups, scaffolding, visuals, real world problems, questioning

techniques and modification of vocabulary (DSM). These strategies were all found in the

articles by Garbrick, Kao, Draper, Foster and Orosco.

What I understood from Garbrick’s claims is that repeated reading supports

student engagement and student learning overall. When students read texts over and over

again that begin to understand it more. Also, students must be fully engaged in reading

and content being learned to learn from it. Teachers must reinforce the benefits of

something for students to realize that something will help them in the long run. What I

can do to help my students that struggle with comprehending math word problems is to

have them read the problems more than once or as many times as they need to. I also

know that students must be fully engaged in the problem to fully understand it, so I

should help student engagement by creating real world problem solving problems that

relate to my students. Teachers should also create a student-centered classroom to

promote better student engagement that will provide students with more opportunities to

immerse themselves with the new concepts and new learning (Garbrick).

My understanding of the claim, “a teacher must tap into the a students prior

knowledge” means that a teacher must be able to relate what the students are learning to

the real world and what they have already seen. Most of the time students have been


exposed to in some form what they are learning. My understanding of the claim,

“students need to be presented with the opportunity to excel and succeed,” is that a

teacher needs to give students who struggle with certain concepts a problem that they are

able to do so they do not hit the point of frustration and give up all together. My

understanding of the claim, “a teacher must have a conceptual understanding of a

standard,” its that a teacher must understand, first, why you solve a problem a certain way

before they try and teach it to their students. To teach a concept in math I must

understand what that concept is and understand it fully before I try to teach it. To help

my students who struggle with comprehension of math word problems I must be able to

relate the concept being learned to their life. To do this I can create real world problems

that the students will be able to eventually succeed at solving. I want my students to be

able to succeed because if I give them a problem that is completely impossible for them

to solve than they will hit the point of frustration and give up entirely (Kao).

According to Draper, effective strategies of proficient math learners. Some of the

strategies that they use are for before reading, during reading and after reading. Before

reading a learner should activate prior knowledge, make connections, predict and

question. During reading readers should make connections, predict, question, visualize,

determine what is important, infer, synthesize and monitor comprehension. After reading

readers should evaluate predictions they have made, question, visualize, determine what

is important, infer and synthesize. In order for students to be successful in the math

classroom they must be able to find the meaning of a math problem and look for

approaches to a possible solution. Students must analyze and make conjectures about

information. They need to analyze situations to make connections and plan solutions.


Reading comprehension and writing strategies are parallel to strategies students need to

be mathematically proficient. Much like literacy, students need to self-monitor, evaluate

their progress and ask questions when necessary. They need to be flexible in using

different properties of math operations. They need to move freely and fluently between

equations, verbal descriptions, tables, graphs, etc. I am able to use the strategies for

before reading, during reading, and after reading. It gives me ways of using the strategies

mentioned in the article that I have never seen before. For example, what kind of

questions to ask and what levels those questions are. Also instead of a KWPR chart they

use a KNWS chart. Which includes what information does a student not need from a

word problem. It also gives me excellent math vocabulary sights that I can use to help

my students (Draper).

Comprehension strategies that can be used when solving math problems are draw

a chart, thinking, make a movie (act it out), predict, read and find key words, Imagine,

reread, read carefully, show your work, use tally marks, looking back in the text, draw

conclusions, skip problems and come back to them later, summarize, sound out words,

and look for base words. I can use some of these strategies in my classroom with my

students and see which ones work best for them. I know that my students already feel

comfortable drawing quick pictures, reading and finding key words, and watching a

teacher model. Some of the strategies I can use for math word problems that I have read

from this article are: drawing pictures, rereading, picking out keywords, creating visuals

using manipulatives, and thinking about the problem (Foster).

Orosco’s study showed that intervention facilitated math problem-solving

performance. When the teacher changed the difficult vocabulary that the students did not


understand to words that were not as complex the students were able to solve the math

problems with less struggle. This strategy helped the teacher understand if her students

were struggling with math content or reading comprehension. I can try to use this

intervention in my classroom with my students and see if it is helpful to them. I feel that

the DSM intervention will help my students because I have noticed that my students

struggle with vocabulary in all subjects areas. If I could change some of the words to

words that are easier to understand, then the students would be able to apply the math

content that they have learned better to solve the problem. It will then be easier for me to

analyze the data to see if my students are struggling with math content or reading

comprehension (Orosco).

Teachers must create real world problems for their students so they can connect

math to their own life. Teachers should ask four types of questions during a lesson. The

four types of questions they should ask are starter questions, getting unstuck questions,

questions that check work and questions that promote deeper thinking. Strategies that

teachers can teach students to help with comprehension of word problems are to make

connections, ask questions, infer and visualize mathematics using visual representations,

summarizing and synthesizing by distinguishing important findings from interesting

details. This book has provided me with strategies I can use to teach comprehension of

math word problems. I can show students how to distinguish important facts from

irrelevant information, make connections, ask questions, and create visual

representations. It shows me sample questions to ask at the beginning, to help students

get unstuck, check work, and to think deeper. I can also try to follow the steps to teach


comprehension: model, guided practice, independent practice, and then incorporation

(Siena).

I also found many ways of collecting data for my inquiry. After finding out what

strategies I could use to help my students, I needed to know how to collect data and what

data to collect. After analyzing other teacher inquires I found out some ways to collect

data. I could collect data through weekly reflections, lesson plans, surveys, observations,

interviews, video and student work. Some of the data I also could use are unadapted

word problems, cultural background knowledge, identification of challenging words and

anecdotal notes. The inquires and articles that I found these different types of strategies

were by Garbrick, Kao and Rawhouser.

All these articles helped me form a better understanding of my own inquiry. They

helped me create a process to collect data after giving me examples of different types of

data I should collect. They gave me strategies that I could use with my students, in the

classroom and how to analyze if the strategies worked or not with my students.


Teaching Context

Fun Elementary School is in the Hillsborough County school district whose mission is to

provide an education that enables each student to excel as a successful and responsible

citizen. And their vision is to become the nation’s leader in developing successful

students. Fun Elementary School is located in Tampa, FL and is one of 175 elementary

schools in Hillsborough School District. Fun Elementary school has 621 students grades

pk-5, a 11 to 1 student/teacher ratio, and has 83.4% of its students with free or reduced

lunch. 44% of the students are girls and 56% of the students are boys. 42% of the

students are black, 35.1% of the students are Hispanic, 15.5% are white and the rest of

the students are Asian or other. My specific classroom has a total of 18 students. Out of

the 18 students eight are girls and ten are boys. Eight of my students have been retained

once in either kindergarten or first grade. I have nine English language learners, seven

boys and two girls. My classroom has two white students, five black students, and eleven

Hispanic students.

Student 1:

Never retained

DRA instructional level 8 in 1st quarter

DRA independent level 16 in 3rd quarter

Ongoing struggle in comprehension

English Language Learner (Spanish)

Average 80% on Unit Math Tests

Struggles in writing complete sentences (Does the complete minimum)


Student 2:

Retained in 1st grade

DRA instructional level 4 in 1st quarter

DRA instructional level 14 in 3rd quarter

Ongoing struggle in fluency and comprehension

English Language Learner (Spanish)

Average 65% on Unit Math Tests

Moves back and forth between Mexico and Florida

Can not write complete sentences


Rationale

After being in my classroom for six weeks, I have noticed that many of my students

struggle in math. I first found out which students struggle with comprehension of word

problems, which students struggle with the calculating aspect of word problems and

which ones struggle in both areas. After I found out why the students struggled with

math word problems I decided, I wanted to research comprehension strategies that could

be used when reading math word problems. I want to know what strategies will help

students comprehend math word problems. Students have to solve problems every day

and to do so, they must first understand the information given to them and know what

needs to be solved. It is the teacher’s job and goal is to prepare students for their future

and for these real world experiences.


Question and Sub-Questions

Question:

How can I incorporate comprehensive strategies into my mathematical instruction to help

my students better solve mathematical word problems?

Sub-questions:

What strategies work best for all students?

How do you know that the strategy is helping the students?

In what ways are reading comprehension and mathematical comprehension strategies

linked?

In what ways are reading comprehension and mathematical comprehension strategies

different?


Data Collection

The quantitative data that I collected was my students Istation results, DRAs (quarter 1

and 3), running records, and summative math assessments. Istation is a computer-based

learning system that offers the depth and breadth of curriculum, quality production,

wealth of resources, and measurable results. I analyzed my student’s comprehension

levels and recorded where they were at and the progress they made. I analyzed

comprehension levels of my students because for students to be able to understand a

mathematical word problem after reading the problem they must be able to comprehend

what the problem is saying and what it means. DRA’s are developmental reading

assessments that a teacher administers to find out what a child’s reading capabilities are.

I analyzed the students DRAs because I wanted to know where the students were with

their reading capability. The DRAs told me what level they were reading at, their

fluency, retelling of a story, and comprehension. These are all skills that that a child

would need when reading a mathematical word problem. Running Records are

assessments that assess reading behavior of a student. I analyzed the running records

because I wanted to know where the child was performing with just reading texts. The

running records told me how well the child could read. Running records tell you how

fluent a student is while they are reading (how many words per minute). I analyzed my

student’s math assessments so I could see where my students struggled while taking math

tests. I noticed that the students struggled with the word problems. The students

struggled with determining when to add and when to subtract. That showed me that the

students might have not understood what the problem was asking. The Unit math tests

also showed me that the students struggle with explaining why they did what they did to


solve a problem. The qualitative data that I collected was student’s math journals,

classwork, homework, and anecdotal notes. I wanted to see how well the students were

doing with answering word problems without my assistance. By looking at the math

journals I was able to find out where the students had a misunderstanding. By collecting

their classwork I could also find out where my students were struggling and excelling. I

collected my student’s homework so I could see where my students are with extra

practice at home, on his or her own. I felt that looking at their homework was very useful

because I know that my students do the homework completely on their own. I was able

to see what my students could do with out any help from classroom help. Classroom help

being my CT and I, the word wall words, and anchor charts. I also wanted to analyze

how the students were solving the problems to see if they were understanding the

information that was given in the problem and if they were showing the correct work, not

just guessing. I also interviewed my students. I asked them what strategies they thought

was the most helpful to them and why those strategies were helpful to them. I did this

because I wanted my student’s opinion on what they thought helped them the most. It is

important to ask the students because they know themselves the best.


Data Analysis

As I was using the deductive approach to analyzing my data I started recognizing my

themes for each student. After I went through my data a couple times I went back and

started recording my themes. The themes I have come up with are overall ability,

comprehension, fluency, strategies, math scores (Unit Tests), Real world problems and

survey questions. After I came up with my themes I began my inductive data analysis. I

chose to look at my students overall reading ability because it is important for students to

be able to read and comprehend what they are reading in math because it will help them

solve word problems. Their overall ability includes fluency, letter knowledge, phonemic

awareness, decoding, and comprehension. Both of my students had an ongoing struggle

with comprehension so I decided to look into more data like their DRA’s, running

records, Istation results, and my anecdotal notes that I have collected to see if all the data

showed this struggle and it did. At first, I skipped over fluency because I did not think it

was important. After reading, “Read it Again” by Garbrick, I found out that fluency is

linked to reading comprehension. I thought it was important for me to go back through

my data and see if either of my students struggled with their fluency. I noticed that

student 1 did not show ongoing struggle but student 2 did. I decided to analyze strategies

that students were using and not using because that was the point of my inquiry. I want

to come up with the best strategies for students to use when they were trying to

comprehend and solve mathematical word problems. I began with analyzing students

Unit Math test scores and came to a realization that I should also look through their tests

to see what types of problems they were missing and what strategies they were using

when solving all types of problems. I noticed that both of my students were missing the


word problem questions in the tests. On their tests they would show correct work for the

way they thought they had to solve the problem but the way they thought was not the

correct way. That data showed me that the students were struggling with comprehending

the word problem and finding out what they had to do to solve the problem. After I

looked at the strategies my students were already using to help them solve the problems I

compared them to the strategies recommended in “Comprehension Strategies Applied to

Mathematics” by Draper and noted that some were the same. I also analyzed my

student’s work after solving real world problems. The work I analyzed was classwork,

homework and journal entries. I observed that my students get confused when they try to

figure out what the question is asking them but they show correct work for the way that

they think the problem should be solved. For example, my students would use addition

to solve a problem when they should of used subtraction. After I analyzed all the data

that I have collected I asked both of my students survey questions. For example, I asked,

“What strategy do you think worked best for you? Why? I found the surveys very helpful

because I was able to get my students feedback on what they think works best and what

they thought did not help them or was too hard for them to understand. This gave me a

better insight to better understand my students thinking.

Quantitative:

The quantitative data that I collected was my students DRAs, Comprehension scores, and

running records. I collected the Istation comprehension reports in October to see where

my students were around the time I was starting my inquiry. They were both below level

and comprehension of texts was an ongoing struggle for both students. I also collected 2

DRAs from each student. I collected one DRA that was done in September for both


students and one that was done in December for student 2 and January for student 1.

Looking at these I noticed that the student 1 improved from a level 8 to a level 16 and

student 2 improved from a level 4 to a level 14.

Data Student 1 (Tier 2) Student 2 (Tier 3)Comprehension (Istation) 10/6: Below level (Better

than or equal to 8% of students) ongoing struggle (ability index)

10/6: Below level (Better than or equal to 10% or students) ongoing struggle (ability index)

DRA (1st) 9/5: Reads at a level independent 8Struggles: previewing (developing) and retelling (developing)

9/9: Reads at a instructional level 4Struggles: retelling (emerging) and reflection (developing)

DRA (2nd) 1/6: Reads at a instructional level16Struggles: reflection (instructional) and making connections (instructional)

12/1: Reads at a instructional level 14Struggles: retelling (instructional) and reflection (instructional)

Running Record (1) 10/10: Instructional level 6Running Record (2) 11/7: Independent level 10Running Record (3) 11/18: Instructional level

12

Qualitative:

1. Real World Problems answered in notebooks

Example Word Problem (GCG):

Joey puts his stickers in 5 rows. There are 3 stickers in each row. How many stickers does Joey have? (I read the problem to the students)

Student 1: Drew 5 rows with 3 stickers in each row and got 15

Student 2: Drew 3 rows with 5 stickers in each row and got 15


There were 16 kids waiting in line to see Santa. 18 more kids joined the line. How many kids are waiting in line now? (I read the problem to the students)

Students 1: Added 16 and 18 together and got 34.


Student 2: Added 16 and 18 together and got 34.


Seven hundred seventeen people rode Cheetah Hunt on Friday. Three hundred twenty eight people rode in the afternoon and the rest rode in the morning. How many people rode in the morning?

Student 1: Subtracted 328 from 717 and got 389.

Student 2: Added 717 and 328 and got 935. Had many erase marks. (confused)


On an oak tree there are 282 fewer red ants than black ants. There are six hundred black ants. How many red ants are on the oak tree?



2. Classwork

Appendix A

Appendix B

Appendix C

3. Group Work

4. Weekly Homework

5. Anecdotal Notes: Notes I took about my students working on word problems. I

watched the strategies the students used to help them better understand the problem, how

long it took them and if the students got the problems correct or not.


Claims

1. Students struggle with understanding what the question is asking in math word

problems. When students read a word problem they have trouble deciding what

they should do to solve for the answer because they are not sure what the

question is asking for.

2. When there is an improvement in a student’s comprehension in reading,

comprehension improves in mathematics. When students better understand a

text, they are able to better answer questions about the text.

3. Students prefer using manipulatives when solving mathematical word problems.

Students enjoy and learn best when they can be hands-on as they are learning and

solving problems.

4. Students struggle with deciding which information they do not need to know

when solving mathematical word problems. When students read a word problem

they are not sure what information is not needed and can be taken out of the

equation to solve for an answer.


Implications

1. After what I have observed in my classroom with my students, I can imply that as

students get better at reading the easier it is for them to read math word problems

but are not always able to comprehend what they have read. Just because a

student is able to read words, it does not mean that a student is able to

comprehend what they have read. Comprehension of a text is a skill.

Comprehension is the ability to understand something. So students are supposed

to be able to understand a word problem after reading it. The students have to

know what a question is asking, the information that is already given and the

information that they have to figure out/solve for.

2. I can imply that just because students do not answer a math word problem

correctly, does not mean that they do not know how to apply the math skill being

learned. This means that sometimes students are either not able to read a math

word problem and/or understand what the problem was asking. They were not

able to comprehend what they have read. Students must be able to understand

what a problem is asking for and what information is already given to them before

they can even apply the math skill that is being learned for that unit. If they are

unable to read or comprehend what they are reading then they will not be able to

show if they have mastered the mathematical skill that has been taught.


Future Wonderings

1. What do you do as a teacher if students struggle with both calculation and

comprehension of math word problems?

2. What is the best way to know if a strategy is really helping the student?

3. What do you do if a child can not read the mathematical word problem at all?


References

Draper, Debbie. (2012). Comprehension strategies applied to mathematics. DECD Curriculum Consultant, Northern Adelaide.

Foster, Shannon. (2007). The day math and reading got hitched. Teaching Children Mathematics, Vol. 14, No. 4. Pp. 196-201. National Council of Teachers

of Mathematics

Garbrick, Steve. (2005). Read it again, Sam: Using repeated reading during guided reading instruction to promote fluency, comprehension, and vocabulary.

Retrieved from file:///C:/Users/carlye1/Downloads/Read%20it%20again.pdf

Glenberg, Arthur., Willford, Jonathan., Gibson, Bryan., Goldberg, Andrew., Zhu, Xiaojin. (2012). Improving reading to improve math. Scientific studies of reading. Retrieved from http://www.tandfonline.com/loi/hssr20

Kao, Emily. (2011). Connecting words to numbers. Retrieved from http://www.ed.psu.edu/pds/teacher-inquiry

Orosco, Michael J., Swanson Lee H., O’Connor, Rollanda, Lussier, Cathy. (2011). The effects of dynamic strategic math on English language learners’ word problem solving. Hammill Institute on Disabilities. Retrieved from http://sed.sagepub.com/content/47/2/96

Parker, Emelie. (2003). Reading for emergent readers. Retrieved from http://www.ed.psu.edu/pds/teacher-inquiry

Rawhouser, Julie. (2005). I’m a new teacher, how can I effectively teach math conceptually? Retrieved from http://www.ed.psu.edu/pds/teacher-inquiry

Reed, Diane., Warner, Amy. (2004). Revisiting reader’s workshop. Retrieved from file:///C:/Users/carlye1/Downloads/ReadersWorkshop.pdf

Siena, Maggie. (2009). From reading to math. Published by Math Solutions


http://www.ed.psu.edu/pds/teacher-inquiry



http://www.tandfonline.com/loi/hssr20

Appendix

Appendix A


Appendix B

Appe


ndix C


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