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Comprehending Mathematics
Carlye Carson2nd Grade Resident
Pizzo Elementary [email protected]
Comprehending Mathematics 1
Table of Contents
Literature Review
Teaching Context
Rationale
Question and Sub-Questions
Data Collection
Data Analysis
Claims
Implications
Future Wonderings
References
Appendix
Comprehending Mathematics 2
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
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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
Comprehending Mathematics 4
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
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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.
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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
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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
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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.
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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)
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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
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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.
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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?
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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
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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.
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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
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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
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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
Example Word Problem (GCG):
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.
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Student 2: Added 16 and 18 together and got 34.
Example Word Problem (GCG):
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)
Example Word Problem (GCG):
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?
Student 1: Subtracted 282 from 600 and got 318.
Student 2: Subtracted 282 from 600 and got 318.
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.
Comprehending Mathematics 19
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.
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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.
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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?
Comprehending Mathematics 22
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
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Appendix
Appendix A
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Appendix B
Appe
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ndix C
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