## Microsoft word - mayo_ar_finaledit_aug2007.doc

Connections Between Communication
and Math Abilities
Rachelle Mayo
Math in the Middle Institute Partnership Action Research Project Report Department of Teaching, Learning, and Teacher Education Connections Between Communication and Math Abilities
Abstract

In this action research study of my Class I School’s 5th and 8th grade mathematics, I investigated
My first assertion is that communication seems to help students stay on task, with feed- back on their solutions; it helps them understand where their thinking is off and where they made their mistakes. In student interviews I asked what they learn by watching and listening to others’ Darrin, an 8th grader, for instance, responds in a way that seems quite normal for him,
How they did it, or how to do it if I don’t know how to do it, and other ways to do
it.
” He has never seemed very confident and does not think of the idea that his
explanations may help others:
Sam responded similarly, “I can see the way they do it and compare mine to it to see
how they did it differently and see if there is better way to do it so I can improve
myself.”
He is always one to find the best way to do anything. And he will always
challenge new information.

Alternatively, Linda expressed that listening and responding to others is not always a
good thing. She said, “Sometimes they can be confusing because they learned
differently, different than the way I have. So it’s a different explanation.”
Linda
indicates that this sometimes confuses her and loses what she thought she had come to
understand.

Amber realized another importance to group work: “There are so many different ways
people see different things.”
Once students understand there is not always one way or
one answer they will be more willing to try to solve problems because they will know
that they can come up with their own way to solve.

Natalie also saw the importance of group work: “They (group members) help me see a
new perspective and I can better understand what I am doing.”
Unlike Linda, Natalie sees the good in being able to hearing different explanations.
–1 = 2 +b I moved the 2 to the other side of the equal by adding instead of
subtracting. Which then gave me the incorrect b so my final solution was wrong too.
I had y = ½ x + 1 and I should have had b = -3 and my final should have been y = ½
x – 3. So be careful with your signs they make a big difference.”

Other students talked about other benefits to justifying their answers. Taryn said: “I can explain it to others or I can do more problems like that one.”
When asked if they can explain a problem to a student who was absent from math class, students verified the benefit to justifying their solutions. Linda wasn’t confident in herself, but knew she could help if she did the justifying on her
own work. “If it was a complicated problem and I had it written down I could
explain it to them.”
Sam stated that justifying can help you as well as others. “If you want to help someone
else eventually then it is easier to show how you got the answer, if you just have an

On April 4th I had the students start a project of sending a problem and a solution to another classroom by Internet. They in turn solved the problem and returned a solution; sometimes a comment of the differences and similarities of how both students solved the problem was included. Some students explained each step while others assumed an easy concept can be skipped over without showing their work. One of the 8th graders felt her explanation of a solution in written words would never end. She had a great detailed explanation. Responses to the sent solutions were just as educational as the solution process itself. Students were able to see other ways of solving the same problem as well as read comments from other students on their feelings and understanding of the sent solution. Some of the 8th grade student from East Butler responses included noticed mistakes: The way the other student did it makes sense to me and I could follow every step, I
found my mistake and understand what I did wrong.

Some of the responses showed an understanding of how the problem may have been done a different way that may have been less time consuming: For the problem, Which point belongs to the graph of the solution set of the system? x<2
and y< 2x +3 a. (0, 5) b. (0, -5 ) c. (-5, 0) d. (-3, 5)
I did a lot more work than the other student, I did not think of plugging them in to
see which one worked. That would have saved me time. I am happy that I got the
same answer after all that work
.
Other responses showed that even though the solution was different the solution was written in a For the problem, 3a + b = 4 and a- 2b = 6 I chose to solve it a different way, by substitution. I could follow along exactly with
what the other student wrote. They did a good job of explaining what to do. They

ended up with the same solution that I got, but solved with a different method,

Some solution responses just showed another way to solve the problem that comes up the same For the problem, The sum of 3 and y multiplied by 3 is less than the sum of 5 and y
multiplied by 5.
After reading the other students solution, I found my solution to be very similar. I
moved the y to a different spot to begin with, but then we ended up with the same
solution in the end.

There were also responses that showed that the solutions sent may not have been completely understood. But were still able to teach mistakes that can be made easily: For the problem, 15 s cubed t over 3 s squared t cubed Sam wrote: First I started of by taking 15 divided by 3. Then I worked with the s
variables. The s cubed need to be divided by the s squared. When there are
exponents though you have to actually subtract them instead of dividing them. Then
on the t exponents it ended up being a negative exponent. The numerator was only t
and the denominator was t cubed so it made a negative exponent. To make it
positive you have to put one over t squared. Then you have to multiply the other
part of the equation into one over t squared. When you multiply them together, the
first part of the solution goes over the t squared because they are whole numbers.
So, you end up with five s over t square.

East Butler Response: I could follow what the student wrote for a solution, but I
forgot about the negative exponents. I was a little confused by the last part. I just
remembered the fact that the variable goes where the higher power was. I did end
up with the same answer though. She was more detailed than me in the response.
Abby’s first experience with a response was not as promising. The problem once again
dealt with money. Abby sent her solution but when the response came back the other student’s answer was not the same. Abby immediately felt she had done something wrong. However she was correct and the other student was wrong. They had forgotten to use decimal and dollar sign. This showed Abby she needs confidence and to evaluate her own and other students’ reasoning and solutions. Abby does have a struggle with self confidence. Taken from the student interview: When asked: What do you think about when your teacher asks questions during math
class?
Abby answered: “If a teacher asks a question I think I did something wrong.”

I have seen improvement in Abby’s solutions. She showed desire to do right and
explained each step with detail and care. For the problem, Sara has \$70 and a \$10 off coupon. If she goes to the Running Place and
buys track shoes that cost \$49.95, a stop watch that costs \$24.99, and two packages of
socks at \$4.99 each, does she have enough money?
Abby’ solution: I added \$49.95, \$24.99, and \$4.99 x 2 which totaled to \$84.92. Then
I took away \$10 for the coupon leaving \$74.92. She only had \$70 so she doesn’t have
Another of Abby’s problems included buying 2 patches at \$2.50 each, 3 magnets at \$2.00
each, a cap at \$8.00, 2 flags at \$3.25 each, and 2 quill pens at \$1.50 each at a museum gift
shop. How much would you have left out of \$28.90 if you bought all the items listed.
Abby’s solution: I took \$2.50 x 2 = \$5.00 in patches. Then \$2.00 x 3 = \$6.00 in
magnets. Then, \$8.00 for a cap. Then I took \$3.25 x 2 = \$6.50 in flags. And finally
\$1.50 x 2 = \$3.00 in quill pens. Now add them together. \$5.00 + \$6.00 + \$8.00 + \$6.50
+ \$3.00 = \$28.50. We spent \$28.50 in all and we only have \$28.90. I took away \$28.50
from \$28.90 and got .40. So we only have \$.40 left over.

My third assertion is that group discussion helps create a self motivated responsibility in the students. Communication between teacher and student helps student get started but student to student communication helps keep motivation strong. The pretests given before each unit were scary for the students. Not knowing how to do something on a test is hard for students to deal with. I did observe an increase in desire to learn the content from the pretest when it came time to learn that skill. I had a hard time too during the pretests. I wanted to help too much. Then during the lessons we all had self motivation to make sure the content was learned well enough for the post test. This teacher student communication was a great start to the motivation but the student to student communication enhanced the motivation and strengthened their abilities. After the first pre-test, post test cycle the students seemed to stretch their abilities to complete the following pre-tests. They weren’t afraid to try any skill they learned previously to complete at least portions of the pre-test. The following was taken from a journal entry during the week of April 11th : The pretest was a good experience; the students were able to use what they know to figure out something new. After each pre-test the student to student communication was exciting. They talked of the processes they tried and were excited to learn if there are easier techniques to solving those types Achievement scores showed improvements as well. As a class the Math Total score increased 9.8% (from a class average of 77.3% to 87.1%). The 5th grader herself increased her Math Total 27% (from 66% to 93%). One of the 8th graders increased their Math Total by 33%. (from 39% to 72%). 72% was the lowest percentile for all the students in the Math Total area There was an obvious increase in the Problem solving portion of the test as well. The class increased by 8% (from 80% to 88%). The same 8th grader that had a Math total increase also had a 22% increase in the problem solving portion (from 50% to 72%). The 5th grader, too, showed and increase in the problem solving portion, 10% (from 79% to 89%). Once again 72% was the lowest percentile for the Problem Solving portion of the test putting all students above Conclusions
All this data supports the literature that I read about the importance of communication. Students are more influenced by communication they experience (Smiley, 1958). Communication including writing enhances the learning of mathematics; it extends their thinking and understanding. In turn engagement in thinking leads to a better understanding and improves communication skills (Stein, Grover, & Henningsen, 1996). Through communication the students are able to clarify their own thinking and make sense of others’ explanations. With an increase in the understanding of mathematics connections between prior knowledge and concepts being taught are being made and are more meaningful. Even if students do not have classmates to communicate with writing solutions and sharing electronically may be a type of communication that will enhance their mathematical thinking. I feel a continuation and even increase in this written communication with grade level students would continue to increase math abilities as well as other subject areas. Although, across grade level solution presentations, did not seem profitable I do however feel it was a step to the written solutions by Internet that Searching for and finding a willing classroom and teacher may be the hardest part of incorporating this into a classroom. Today’s students enjoy using computers and socializing with other students. Using this interest can help increase math abilities. This data also shows that group work can generate more ideas for solutions. Students know that everyone has different ways of doing things. Working in a group can bring about more and even creative ideas for solutions. Exposure to these different ways can create an increase in Increases in Achievement scores may not be the result of only this research but other factors as well. Natural intellectual growth may have caused an increase as well. Other factors such as a change in school settings in the coming year may have placed an added incentive to do their best on the Achievement tests to prove their highest abilities. Due to a closure of the Class 1 schools my district will close the site at which I am at. All students will transfer to other districts. This will give classmates to the one 5th grader. The 8th graders will have a transition into a high school setting which will be very different from there past experiences. Implications
I will be in a larger school district next year in the 5th grade level. However, I will not be teaching math because of departmentalizing. As a result of this research I will try to encourage my new school district to permit me to continue the Internet communications. I can inform them of the improvement I observed in the students performances as well as their scores. I can ask to use the Internet communications in other subject areas to see if this written communication increases other subject areas as well. I will however try to keep up with my students’ math education, by having them write up solutions to one problem a week to inform me of the content they are learning. This can help me keep their content areas connected. I hope to excite the teachers in my new district enough to have them try the process of Internet communication in their classrooms. We can, as a team, keep collecting data to see if the technique works in larger school settings and other subject areas. My advice to other teachers with the problem of a lack of communication at a grade level is to find an equivalent classroom willing to communicate by Internet. Then share solutions slowly at first so it is not overwhelming. Many students will enjoy the process enough to do it on their own once we show them how. Class discussions should also be increased in classrooms. Discussions may be time consuming but are very beneficial to you and the students. You can see and hear their understanding and students can learn from each other when they do not quite I hope to follow up on the present seven students to see if their math scores increase or at least stay above average. Due to my closeness of the families this should be obtainable for me. References
Clarke, D., Waywood, A., & Stephens, M. (1993). Probing the structure of mathematical writing. Educational Studies in Mathematics, 25(3), 235- 250. Lappan, G. & Ferrini-Mundy, J. (1993). Knowing and doing mathematics: A new vision for middle grades students. The Elementary School Journal, 93(5), 625-641. Mathematical Association of America. (1991). A call for change: Recommendations for the mathematical preparation of teachers of mathematics. Washington, D.C.: Author. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. Shield, M. & Galbraith, P. (1998). The analysis of student expository writing in mathematics. Educational Studies in Mathematics, 36 (1), 29-52. Smiley, M. (1958). Do your classroom procedures really teach communication? The English Stein, M., Grover, B., & Hennigsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455- 488. Taylor, R. (1989). The potential of small-group mathematics instruction in grades four through six. The Elementary School Journal, 89(5), 633-642. Yackel, E., Cobb, P., & Wood, T. (1993). Developing a basis for mathematical communication within small groups. Journal for Research in Mathematics Education, 6, 33- 44. Yackel, E., Cobb, P., & Wood, T. (1993). The relationship of individual children’s mathematical conceptual development to small-group interaction. Journal for Research in Mathematics Education, 6, 45- 54. Student Interview Questions