Writing is a natural process, a method of communication between people and a way to express the thoughts and feelings that occur within a person. Its use as a tool for the teaching and learning of mathematics is a recent development, springing in part from the NCTM Standards on Communication. No longer the exclusive province of the humanities, writing is now in use in mathematics classes at all levels, K-12.
Through the use of writing in the mathematics classroom, students can clarify their thinking, recognize and appreciate the connection between mathematics and other disciplines, and communicate their thoughts, ideas, and understanding about the subject with other students. Writing involves all students, both male and female, and the teacher. It is important for the students to see the teacher participating in these writing activities. Writing provides an alternative mode of learning for those creative students who have not previously been reached by more traditional, structured, linear methodology. Because a student often knows more than he or she can explained verbally, writing helps the student to uncover more of what is known and to express it. Once the students have had the opportunity to collect their thoughts on paper, they may then volunteer to share their work.
Finally, as Joan Countryman says in her book, Writing to Learn Mathematics:
Mathematics is a way to understand the world and writing is a way to understand mathematics.
We agree with and endorse these statements.
From Joan Countryman's book, and our own ideas, we came up with a list.
How can we bring writing into the mathematics classroom effectively?
Writing activities can be used BEFORE a lesson, during a lesson, and after a lesson. Often writing activities used before a lesson help the student to activate the prior knowledge he/she has on a particular subject. By having the student write these ideas they see what they know. After ideas have been written, students can then share their ideas with others in the class. These ideas are the result of free-writing and should be considered tentative. Students are just trying to get their thoughts down on paper. Through the lesson activities students will be able to clarify some of their tentative thinking expressed in the BEFORE writing. Many BEFORE writing activities involve students making lists, writing questions they have about a topic, brainstorming associations, mapping relationships, or writing a personal narrative. Several of the activities mentioned in this module can be used as starter activities.
Writing can also be used DURING a lesson. When a class has been learning a new idea, it might be appropriate to stop the lesson and have the students write a short paragraph describing what their understanding of the material is or possibly apply the newly learned idea to a new function and explain what happens. Then each student can be paired with other students and read their summaries to each other. Volunteers can read their responses to the entire class. Through this process the teacher can assess the class's understanding of the new idea. The thinking and writing that takes place here is still somewhat tentative. Once this is done, the class returns to additional instruction and investigation where they can continue to refine their thoughts. DURING class writing can involve students in asking questions, noting difficulties they are encountering with new material, or again mapping relationships.
Writing activities can also be used AFTER a lesson. These types of activities are probably more commonly known and used than BEFORE and DURING activities. Journal writing is a very common technique used for AFTER writing activities. Students are asked to reflect back to the lesson. The thinking and writing, at this stage, is much less tentative. Some journal writing might ask students to retell an idea or write an idea from a different perspective or point of view, to write questions they have about a lesson, or to prevent a logical solution to a problem. (Lytle, 1988)
A word web is an exercise in free association which is useful for opening a channel of creativity and for getting thoughts flowing. It can be done either as a self directed activity or as a teacher directed one. ( An example is shown on the next page.)
Step One: In the center of a blank piece of paper the person constructing the web writes one word and circles it. A circle is used for this center word because the circle is basic to all human endeavor. It is the first geometric shape drawn by young children in all cultures. It is the basic shape of the social structure in all cultures, including primitive ones (example: a circle of story tellers around an evening fire).
Step Two: For five to ten minutes after writing the center word and circling it, the person making the web free associates, writing other words and phrases that come to mind. No attempt is made to make sense of the connections that come to mind. Each new word or phrase is connected--either to the original, circled word or to one of the subsequent ones.
New words can be connected by arrows, segments, or whatever comes to mind. They can be circled, enclosed within other shapes, written along an arrow, or whatever the writer desires.
(Note: if students are working on their word web in a classroom setting, it should be done in silence and the time period to be allotted for this activity should be stated at the outset. The teacher , also, should write during this time.)
Step Three: At the end of the free association time period, the person making the word web selects one or more of the words and phrases that he/she has written and writes from that. This second writing can be a paragraph, a poem, a letter, or whatever has been decided upon.
The KWL Activity can be used in conjunction with any unit of material across the curriculum. It is made up of three parts: K, what the student knows, W, what the student wants to know, and L, what the student has learned. One way to use this activity is to place three columns each labeled K, W, or L, on a sheet of paper. At the start of a unit distribute the paper and ask students to complete the first two columns. They write down all the things they know about a particular subject under the K column. This could include just a list of terms, formulas, symbols, concepts, and/or relationships between concepts. Under the second column the students write down a list of ideas they WANT to know about. Students can list ideas or relationships they are not sure of or ideas about which they have questions. At this point a teacher could conduct a class survey or just peruse the data to determine the prior knowledge of the class on a particular subject and where the class's interest falls on a particular topic. It is also possible to have students select one or two topics and write a more detailed description of what they know or what they want to know. The activity is then set aside until the end of the unit . At the end of the unit the students are asked to complete column three and tell about those things they have learned through their work on the unit. Students should be encouraged to refer back to the K and W columns when they are completing the last column. Upon completion of the entire activity, the student and teacher can both look back and see the growth each student has made. This activity can also be used for a one day activity. Students would complete the L at the conclusion of the lesson.
Students are asked to write a short letter to a younger student. For example, a Calculus student might write to a Precalculus student; an Algebra II student could write to an Algebra I student, etc. The teacher can specify, saying, for example " Write a letter to John, who is having trouble factoring in Algebra I. Explain to him how you decide which method of factoring to use." These letters can be included in the student's journal and may be shared with others in a small group setting or in the class as a whole if the author feels comfortable with this.
This activity can be used in any level class, in any course, and at any time during the year. Each pair of students is given two identical baggies of gumdrops and toothpicks and a small lunch bag. Each baggie contains a given number of toothpicks and a set of colored gumdrops. The pair of students decide on a specific two- or three-dimensional structure to build with their toothpicks. As the students are building the structure they talk about and record a set of directions that another pair of students can follow to reproduce the structure. Upon completing the structure, the pair of students place their construction in a lunch bag. Another pair of students then take the extra baggie of toothpicks and gumdrops and try to follow the directions to build an exact copy of the structure built by the first pair of students. Upon completing the second model the students compare their model with the original model stored in the lunch bag. Students enjoy this activity. Students discover they use a large number of mathematical terms to give clear directions. A teacher can use this activity to survey the mathematical vocabulary which students use as part of their directions. The teacher may decide to make a collection of the key words used by the students to build good directions and build a lesson on these key words.
Materials needed: 1 pair of scissors, 1 piece of paper per student
This cooperative learning group activity uses verbal, writing and listening skills culminating in each student's having a personally made set of tangrams. The students are divided into groups of four. One student, the speaker , faces the teacher so he/she can clearly observe the teacher's actions. Another student, the writer, is to write down exactly what the speaker says. Other members of the group are seated so they cannot see the teacher. As the teacher performs certain actions, the speaker gives verbal instructions to the other members of the group so they can duplicate the actions of the teacher.
When the teacher has completed the series of actions to be performed, the group members who have been following the speaker's instructions should now have made a complete set of tangrams. The roles are now reversed. One of the group members, who has made a tangram set, takes the paper the writer has written and literally, word for word, reads back what the writer has written. The previous speaker and writer now follow these instructions and attempt to make their own tangram set.
Action by teacher:
Make a square from the rectangular piece of paper you are given by folding down a corner; then crease the rectangle below the square. Cut off the rectangle; discard it.
Cut on the diagonal of the square so two triangles are formed.
Hold up 1 of the large triangles. Fold it into two congruent isosceles right triangles.
Cut on this fold, and set these two smaller triangles down.
Hold up the other large right triangle, hold the right angle in one hand, bring that vertex to the hypotenuse and fold; cut on fold.
You now have a trapezoid in your hand. Fold it in half; cut on fold.
You now have two right trapezoids. Hold one, fold it to form a square and a triangle. Cut on that fold.
Pick up the last piece (the other right trapezoid) and fold it so you form a triangle and a parallelogram. Cut on that fold. (That's the tricky fold...practice first!).
The teacher may pause (after each step is completed!) to verbalize: "I heard some good mathematical terms being used...fold on the diagonal of a square...". At the culmination of this activity, each student has her/his own set of tangrams (for future use). The need to communicate clearly becomes readily apparent through this exercise. This procedure can be used with tasks other than making a set of tangrams.
This is a very simple activity to use when you sense that the students do not understand the lesson. Stop, and have them explain what is giving them trouble. After having them "free-write" for a couple of minutes, they should share their writing with their neighbors. This can help clarify their confusion with the material.
When faced with an assignment to do a project in mathematics, some students typically will select to make something ( a model or display, for example), while others will want to write a report. These reports tend to be cut-and-dried, dealing with a famous mathematician or a well-known theorem. Preparing them is seldom pleasurable for the student . Reading them is a chore for the teacher; grading them is even more so. An alternative to this type of report is a less structured one which involves mathematics and some other discipline. These are independent investigations, which may be accompanied by graphs, charts, photographs, or drawings. Listed (below) are some suggestions for Linking topics.
Mathematics and Creative Writing: Write a story. For example, tell the story of someone learning mathematics: a young child, a student coming to America from another country, a person from another planet.
Mathematics and Architecture/Art: A collection of photographs of local architecture, showing structures which incorporate geometric shapes or which employ mathematical concepts; or a collection of photocopies of works of art with the same features. These should be accompanied with explanatory text.
Mathematics and Psychology: A collection of optical illusions with accompanying explanatory text.
Mathematics and Drama: An original skit or one act play, written by a single student or a group of students and performed before the class. Good topics for the play include lives of famous mathematicians or important discoveries.
Although this technique can be used in any mathematics class, it seems to work especially well in a Calculus class. A student is assigned to keep the log for a specified period of time (a day or a week, for example). He/she then keeps notes as usual, but in the log rather than in his/her own notebook. The log is kept in the classroom, to be used by students who have missed class, or as a point of referral for anyone. Some teachers use a spiral notebook for this purpose; others provide carbon paper and typing paper. The advantage of a notebook is that is has everything together. The advantage of the use of carbon paper is that the student taking the notes has a second copy for personal use.
Having students write a children's storybook is a wonderful activity which gives students an opportunity to be creative and, at the same time, review some of key topics they have learned during the year.
To begin, it is a good idea to visit the local library in your community and make a collection of some children's books on the subject of mathematics. Read these books to the class and talk about qualities that make a good children's book. In a geometry class students recently had the task of writing a children's story about a new idea they had learned in 10th grade geometry so that a child could understand the concept. Students are encouraged to use visual representations to enhance their story.
Students write their stories on folded 8.5 x 14 paper which is then bound using cardboard, contact paper, a long necked stapler, glue, and an inside cover sheet.
If it is possible it is a good idea to have the students visit a local elementary school and read their stories to some of the younger students in the school.
Librarians are a wonderful resource for this activity.
Journals have been used across the curriculum, but very recently many teachers have started using them in mathematics classes. It is a good idea to plan some varied writing prompts that will help the student address some of the concepts covered in a lesson.
One idea is to use a WORDBANK. In a WORDBANK students are given a list of at least 4 words (3 of which are easily related and one which is slightly related). Students are asked to write a meaningful paragraph relating the 4 words. Students should not write definitions of the 4 words.
An example of a wordbank journal entry: Use the following set of words to describe what you have learned about the Pythagorean Theorem: legs, triangle, hypotenuse, right, acute, obtuse.
WRITING QUESTIONS A journal entry could be to have students write three "Jeopardy" type questions (along with their answers) or three questions they would like asked on the next test. This gives the students an opportunity to make connections on their own and to think about the type of question that could be asked.
POINT OF VIEW In this type of entry the student is to think of him or herself as possibly a rhombus. The students think and then write about the ways people mix them up with other polygons. This type of activity can be used to assess a student's understanding of a key vocabulary word.
CARTOON Have students draw and write a cartoon about a current idea you are covering. This type of activity gives students who are artistically talented a chance to use their skills within a mathematics class. Other students find that they can easily make stick figures out of numbers, symbols, and figures. Another idea would be to take an appropriate commercial cartoon, white out the dialogue bubble and give students an opportunity to add their own narrative to the cartoon.
REWORDING A DEFINITION OR THEOREM Students learn by taking a difficult theorem or definition and rewriting the theorem or definition in their own words. Teachers can also ask a student who is having difficulty understanding a given statement to begin by making a diagram of what the statement is saying. You could also ask students to make a list of things the theorem does not say. This often helps understand why certain phrases were used in the statement.
Other ideas which "foster the development of critical thinking skills while reinforcing concepts from the mathematics curriculum" (Glatzer, 1993) are found in two books by Dave and Joyce Glatzer. (Math Connections and Math Intersections). The ideas presented in these books can be easily adapted to any level of mathematics class. Here are some examples:
HOW DO YOU KNOW THAT? Write one or more complete sentences to answer each question.
SOMETIMES TRUE - SOMETIMES FALSE Each condition below is true in some cases and false in at least one instance. For each item, give one example that shows when the condition is true and one that shows when it is false. Your answer may take the form of a diagram, an expression, an equation, or a written response.
WHAT ARE YOU LIKELY TO BE ASKED? For each exercise, write one or more questions that you are likely to be asked about the given condition.
MAKING THE CONNECTION What is the obvious conclusion for each of the following? Explain why your conclusion must be true.
Algebra: The trinomial
A final note on journals. Having students go back and revisit their writing is a worthwhile experience. You could have them pick out their favorite journal entries and then write a table of contents, an introduction, and then a conclusion. This will give their writing a nice summary and purpose.
A portfolio is a collection of a student's work that represents her/his mathematical learning throughout high school. She/he is to have TOTAL ownership of the portfolio. It is meant to be a reflection of the depth and breadth of the student's mathematical creativity and understanding.
Students can share what they have learned through adding narratives to the work they include in their portfolios. Some of the writing previously mentioned was tentative, but the writing placed in the portfolios includes ideas students have dealt with earlier and now the ideas should be less tentative. The writing enclosed in a portfolio is used to assess a student's understanding of new mathematical ideas. The contents may be edited, clarified, revised, deleted or added to as a student deems appropriate. Others may assist in this process but only to the extent of aiding the student in re-assessing her/his own work, by questioning, critiquing, suggesting and guiding. "The portfolio should reflect the extent to which a student has gained mathematical power (this includes the development of personal self-confidence,"...NCTM Standards.
Included in the portfolio should be a variety of problem-solving strategies used to solve meaningful problems, entries that evidence a clear understanding of the mathematics and mathematical processes used, and explorations that display an integrated view of the relationships of mathematics with other disciplines and with the real world.
A rubric for portfolio assessment could include a completed table of contents, a letter written by the student to the reviewer describing her/his portfolio, and a holistic scoring rubric indicating level of proficiency. Several state departments of education have developed rubrics for evaluating portfolios.
Tests are an excellent place for students to be given opportunities to express themselves through writing. Students can be asked to make connections to several ideas learned in a unit. They can be asked to make a list of ideas they have learned and then pick two or three of the ideas and write extensively about them.
If a student is given a multiple choice type question, the teacher could ask the student to defend why they have chosen one choice over another. Or a teacher could pose a type of question which would involve a calculator investigation and the student would respond by describing the type of approach he/she used to solve the problem.
Writing a cinquain is one way to give students a format to express their ideas on learning a new idea or mathematical relationship. Students write a poem following this pattern.
How do we assess the writing that our students do?
Here are some helpful hints:
This is only the beginning. Writing in the mathematics classroom takes commitment and prior planning. Hopefully, you can see the benefits of writing as a valuable teaching tool. Once you get started, share your ideas and talk to others. This will keep the exercise exciting and fresh for you and your students.
Countryman, Joan. Writing to Learn Mathematics. Heinemann Publications. 1992.
NCTM. Curriculum and Evaluation Standards for School Mathematics. 1989