Saturday, September 12, 2009

I do and I understand, I reflect and I improve. (writing in mathematics education).

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Incorporating writing into the study and teaching of mathematics can drastically improve teacher understanding of student difficulties with the subject. A program through which education majors are encouraged to get student feedback and solutions in writing is described. This method successfully reveals the level of student knowledge and understanding of mathematical topics. These responses help mathematics teachers and students to reflect on their mathematics teaching and learning processes, respectively.

Full Text :COPYRIGHT 1994 National Council of Teachers of Mathematics, Inc.
Students' writing tells us more than viewing single answers on worksheets

"I hear and I forget. I see and I remember. I do and I understand." In my college methods course on teaching and learning mathematics, my goal is to prepare prospective elementary teachers to meet the challenge of implementing the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). My colleagues agree that it is important for our students majoring in education to develop understanding by "doing," so our students are given the opportunity to plan and teach lessons in a clinical classroom during the semester in which they take their methods courses.

When I observed my students teaching in their classrooms, I was pleased to see them using manipulative materials in activity-based mathematics lessons. But I noticed a glaring omission. Even though they had heard me emphasize the importance of integrating writing with mathematics, and they had seen me exhibit examples of children's writing, the majority still were not using writing as a tool for teaching mathematics. Something had to change. In designing my next course, I restructured the requirements to help them appreciate the value of making writing a regular part of the mathematics curriculum. To ensure that they actually experience writing, my students are required to incorporate the activity into at least one mathematics lesson that they teach in their clinical classroom and to reflect on how the information they gained from this experience could be used to improve future lessons.

Reflective Practice

Teaching a mathematics lesson in which they integrated writing was an essential first step for my students, but the crucial component was reflection. NCTM's Professional Standards for Teaching Mathematics suggests that preservice teachers of mathematics should have opportunities to "analyze and evaluate the appropriateness and effectiveness of their teaching," (NCTM 1991, 160). After teaching their lesson, my students were asked to consider whether their intended objective for having the children write was met, what insights into the children they gained from the experience, and how this information could be used to improve instruction. They submitted these reflections to me along with the children's writing.

In a recent Arithmetic Teacher article, Hart and her colleagues encourage teachers to become reflective practitioners and use the information they gain "to guide and change their instructional practices so they can be more effective" (Hart et al. 1992, 40). Reflection is a powerful change agent because it enables teachers to recognize how their actions affect the classroom environment. By reflecting on the teaching and learning process, they gain insight into how better to meet the instructional needs of their pupils.

Incorporating Writing into Mathematics

To familiarize my students with the numerous ways to incorporate writing into mathematics, I suggested that they read Countryman's (1992) Writing to Learn Mathematics. In addition to describing various writing activities, her book indicates several reasons for having children write about mathematics. By forcing a slowdown in the thought process, writing enables the mind to clarify ideas and more fully integrate new knowledge. Each child is actively involved in reflecting on what she or he has been doing and thus has the opportunity to formulate and rethink her or his ideas. Learning becomes an active process of knowledge construction and sense-making by the student. This writing also helps the teacher evaluate students' understanding and thus make better instructional decisions.

Several of my students asked their pupils to write how they felt about mathematics and what they liked or did not like about learning it. Typical responses from children who disliked mathematics were similar to those of a fifth-grade boy who wrote, "I hate math because it is my worst subject. I do bad in it. I do not understand some things." Reflecting on the children's replies, my students gained insight into the importance of making assessment an integral part of teaching. As one of my students wrote, "No wonder some kids don't like math! It's up to me to continually check and make sure that each child understands and is successful." When asked what he liked most and least about mathematics, a sixth-grade boy wrote, "My favorite part is when we get out of class. The hardest part is remembering all the stuff." My student observed, "These kids view math as a collection of arbitrary rules to memorize. I need to help them see that math makes sense and get them actively involved in the learning process. If we fail to build conceptual understanding and only teach children how to do the procedures by rote memory, many are going to be confused and alienated."

Sometimes we learn as much about ourselves and our teaching from the children's responses as we do about them. When asked to write "what math means to you," almost every child in a fourth-grade classroom identified mathematics with number: "Math is numbers, decimal points, adding, subtracting, multiplying, and all that stuff." The NCTM's Curriculum and Evaluation Standards (1989) recommends that computation be only one facet of the mathematics program, but for these children, as shown by their written responses, computation is the entire program. In her reflection, my student commented, "I want to show these kids that math is far more than pencil-and-paper computation. I want them to experience the excitement of data collection, the beauty of geometry, the exhilaration of solving a nontraditional problem."

Children's writing can give much more insight into what they actually know about a topic than can their responses on a worksheet. One of my students was teaching a fraction unit in a first-grade classroom. To evaluate the children's understanding of the concept, he asked them to write about and depict a situation in which they might use fractions in everyday life. Their pictures showed whether the child could represent the fraction accurately. To facilitate reading, some spelling and punctuation of the children's texts have been corrected. Jared wrote, "Me and my friend are eating pizza at my house and having fun together, and we divided it into eight pieces"; he drew eight equal-sized pieces. Katie's reply, "I share a bar of chocolate with my friend. I cut the chocolate bar with a knife. We each got 15," showed that she realized that 15/30 was equivalent to 1/2, but the thirty pieces in her drawing are not equal in size. This information allowed for more individualized subsequent instruction. As my student commented, "I'll need to do some follow-up work with Katie to make sure she understands that fractional parts have to be the same size." He also discovered the value of an open-ended assignment, which enables each child to contribute. "All of the children were able to think of something in their lives that gets divided into equal parts and to write about it. Everyone was successful. If I had just given them a worksheet, some of the less capable students might have struggled."

Several of my students learned through experience that the writing assignment needs to be structured carefully if it is to yield useful information. To pretest her fourth graders' knowledge before a unit on geometry, one student asked her class to define geometry. Greg wrote, "Geometry is different kinds of shapes like squares and stuff." Sarah's trusting answer was, "I don't know what it is, but if you say it is fun, then I'm sure I'll like it." My student found that it was "hard to gauge from their responses how much they actually knew about geometry" and concluded that if she were to try this activity again, she would need to ask specific questions about the geometric concepts she intended to teach.

By asking more focused questions, another student found writing to be a valuable diagnostic tool in evaluating her third graders' understanding of multiplication. As seen from his explanation, Brian understood multiplication as repeated addition but viewed "4 X 8" as eight sets of four rather than four sets of eight. My student planned further instruction for Brian to have him use the more accepted convention, as well as a review of the array model of multiplication, which he did not mention. Although most of her class understood the process of multiplication, their answers to "When do you use multiplication?" showed that they were unable to apply the concept in real life. Brian was prompted to use the operation only when he saw the multiplication sign in a problem. Many other students replied that multiplication is used "when you have a worksheet to do" or "in math." My student observed, "[M]ost of the children don't know when they would use multiplication other than in math class. This showed me that I need to incorporate lots of real-life examples into my lessons."

To help her children see this connection between the mathematics they learn in school and its application to everyday life, another of my students asked her third graders to write story problems that used multiplication. The children were all successful, but most were not as fanciful as Dan, who wrote, "I went to a Christmas party, and there was a mistletoe right by the door. I was kissed by 16 girls 2 times each." My student reflected, "This lesson helped the children bridge the gap and apply mathematics to their own personal lives. It is evident from their stories that they know how to use multiplication in a meaningful way."

When another student discovered that her fifth graders were "confused about geometry and didn't really like it," she had them cut from magazines pictures of rays, points, planes, and line segments. Then they wrote about what they had learned. She hoped to "give them a better understanding of the terms they were studying, and a better attitude toward geometry." From the class's responses, it was evident that this objective was met. Kim, whose response was representative, wrote, "Today I learned that a simple fence was a pair of parallel lines! A sky is a plane! And spotlights hitting each other are rays! I think it's really cool, because I didn't know that in my backyard there are lots of lines of geometry! Miss C. makes it fun to do math! I can't wait till tomorrow!" My student, Miss C., reflected, "I gained a lot of insight into the students from this project. They really liked being involved and a part of learning, rather than just taking notes on terms. I think the writing really helped them pinpoint exactly what they had learned, and it allowed me to see what the students understood."


The preservice teachers from my classes said that this project convinced them that incorporating writing into mathematics had many advantages. They saw it as a tool for helping children understand and communicate mathematics. Through writing, the children were able to make sense out of mathematics and recognize its relationship to their everyday lives. Every mathematics class presents opportunities for writing. Writing about how they feel about mathematics helps the students focus on what works or does not work for them. This type of writing gives the teacher insight into the attitudes and needs of the individual student and may uncover mathematics anxiety or parental pressure. When negative attitudes toward mathematics are revealed, teachers can identify the causes and help change them.

At the beginning of a new unit, asking the children to write what they know about the topic and what they think they will learn can be a useful diagnostic tool to determine students' current level of knowledge. This rich information base will allow the teacher to tailor lessons to meet the individual instructional needs of the class. Writing at the beginning of a unit also develops a mindset for studying the topic and helps students form connections between what they already know and the new material.

Open-ended writing as a follow-up to a lesson helps students mentally reconstruct what has gone on during the lesson. The last few minutes of the class period can be used to have the children write about what they do or do not understand about what they did in class. By writing about what they have just learned, they reframe the knowledge into their own words, extending and deepening their understanding. Being able to glimpse the children's thinking through their writing enables teachers to diagnose misconceptions and problems more clearly, correcting errors as they arise.

Many students do not understand routine textbook word problems. Having them formulate their own problems, either individually or as part of a cooperative group, empowers them to connect mathematics with real-life situations that make sense to them.

A student mathematics journal also might be used. This journal can be a collection of free-writing comments about the material, their progress, or the mathematics period in general. To gain specific information, teachers might ask students to complete a sentence stem, such as "I think calculators..." or "To study for a math test, I usually...." Such journals are especially valuable if they become dialogue journals in which the teacher writes replies to the students' entries.

The preservice teachers who participated in this project commented that they believed that one of the biggest assets of having their students write was that it furnished information about the children that could be used to improve teaching. By analyzing their students' needs, they could structure more effective lessons. Reflection helped them to see what was working and what needed to be changed. I hope that when they have classrooms of their own, they will become reflective practitioners who will continue to grow and improve throughout their careers.


Countryman, Joan. Writing to Learn Mathematics. Portsmouth. N.H.: Heinemann Educational Books, 1992.

Hart, Lynn, Karen Schultz, Deborah Najee-ullah, and Linda Nash. "The Role of Reflection in Teaching." Arithmetic Teacher 40 (September 1992):40-42.

National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston Va.: The Council, 1989.

-----. Professional Standards for Teaching Mathematics. Reston, Va.: The Council, 1991.

Carolyn Schiebelhut teaches at Concordia College, Moorhead, MN 56562. She also presents workshops for elementary school teachers on implementing NCTM's Curriculum and Evaluation Standards.

Source Citation:Scheibelhut, Carolyn. "I do and I understand, I reflect and I improve." Teaching Children Mathematics 1.n4 (Dec 1994): 242(5). Academic OneFile. Gale. BROWARD COUNTY LIBRARY. 12 Sept. 2009

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