From Mathematics Teaching in the Middle School 9 (May 2004): 490–93.
Improving Writing Prompts
C R A I G A. S J O B E R G, D A V I D S L A V I T,
AND
TERRY COON
Writing elicits student thinking. Students are forced to reflect on what has been learned. Research on metacognition suggests that thinking about thinking helps to cement ideas and concepts into memory. It also helps to clarify ideas. Writing
in math also strengthens writing in general.
—Terry Coon, sixth-grade mathematics teacher
H
AVE YOU EVER ASKED STUDENTS FOR A WRITTEN
explanation of their thinking or how they solved a problem, and their responses did not meet your expectations? Because the teaching of mathematics continues
to move away from a sole focus on correctness and a
finished product to include a focus on process, context, and
understanding (Miller 1991; NCTM 2000), students need
opportunities to express their ideas. A writing task can be an
ideal tool for supporting this important learning activity.
To this end, the prompts in figure 1 were used successfully in eliciting detailed, clearly written, and mathematically powerful responses from our students. We invite you
to try them with your students.
Note that the first prompt is expository in nature. When
students are asked to explain something, they can relate
this process to similar writing tasks, such as describing
how to cook or construct something. This prompt was in1. Explain at least one thing you learned in mathematics this week.
2. How does your answer to question 1 connect to
other things in mathematics?
3. How might you use this in the real world?
Fig. 1 The prompts used in this activity
CRAIG SJOBERG, csjoberg@egreen.wednet.edu, teaches sixth- and
eighth-grade mathematics at Shahala Middle School in
Vancouver, WA 98683. His interests in mathematics education
include exploring teaching strategies that reach students who have
historically struggled in mathematics. DAVID SLAVIT ,
dslavit@wsu.edu, who teaches at Washington State University,
Vancouver, WA 98686, studies student learning in mathematics,
with an emphasis on algebraic development across the K–12 spectrum. He is also interested in technology use and teacher professional development. TERRY COON, tcoon@egreen.wednet.edu,
teaches sixth-grade mathematics at Shahala Middle School. He is
also a math coach who works with mathematics teachers to
improve students’ learning opportunities.
490
tended to develop the connection between mathematical
problem solving and expository writing. Prompts 2 and 3
were selected because of their cognitive nature; specifically, they were to promote students’ use of prior knowledge, to make the connection that mathematical concepts
are built on one another, and to have students reflect on the
mathematics that occurs outside the classroom. The rubric
shown in figure 2 was used to provide expectations for students and assess specific criteria. We found that these
prompts supported our instructional goals and were adaptable to our mathematics curriculum. Qualitative judgments
had to be made in applying the rubric because of the nature
of certain student responses and gaps in the stated criteria.
Development and Use of the Writing Prompts
WE BEGAN OUR ACTION RESEARCH BY POSING THREE
different writing prompts to elicit students’ thinking. Two of
the authors (Craig, a preservice teacher, and Terry, an experienced sixth-grade teacher) used the following prompts during the first half of their one-year, coteaching experience:
1. What? (What have I learned?)
2. So what? (What difference does it make?)
3. Now what? (What can I do with this information?)
The responses to these questions had generally been
disappointing, possibly because the prompts were ambiguous and because most students had not previously been
Edited by JENNY BAY-WILLIAMS, jbay@ksu.edu, who teaches at
Kansas State University, Manhattan, KS 66506
“Take Time for Action” encourages active involvement in research
by teachers as part of their classroom practice. Readers interested
in submitting manuscripts pertaining to this theme should send
them to “Take Time for Action,” MTMS, NCTM, 1906
Association Drive, Reston, VA 20191-1502.
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Copyright © 2004 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use associated with PBS TeacherLine
only. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.
to Improve Student Reflection
0—No response
1—Answer includes mention of mathematical concept
but—
• does not explain concept or convey an understanding through example,
• does not connect to at least one other concept, and
does not provide a real-life example.
2—Answer includes mention of mathematical concept,
explains concept, and conveys understanding through
example but—
• does not provide an example or a connection to at
least one other concept, and
• does not provide a real-life example.
3—Answer includes mention of mathematical concept,
explains concept, conveys understanding through example, and makes a connection to at least one other
concept but—
• does not provide a real-life example.
4—Answer includes mention of mathematical concept,
explains concept, conveys understanding through example, makes a connection to at least one other concept, and provides a real-life example.
Fig. 2 The writing in mathematics scoring rubric
exposed to such writing exercises in mathematics. In addition, Craig and Terry had not adequately modeled or
demonstrated their desired approach to the writing exercise. Because of the poor responses, it was also difficult to
use the reflective writing to adequately assess students’
learning. All three of us agreed that student reflection
could be improved and that writing should be the vehicle
for improvement. It seemed that we were using an approach that we thought to be worthwhile, but we were not
getting the results we hoped for. Should we scrap it?
Change it? Try something new?
Craig decided to create new reflective writing prompts
(stated earlier), which were introduced in the first week of
February. A tiling activity was chosen to model quality responses to the new prompts. Students explored which
shapes among a group of simple polygons could be used to
tile the top of a table. Craig introduced or further developed
ideas of angle measure, shapes, and the term vertex in this
discussion. He also focused on the need for the appropriate
angle measures to add to 360 degrees for a tiling to occur.
He then elicited ideas for possible applications of this task,
and the students discussed patios, ceilings, and floors as
being specific contexts. At the conclusion of the lesson,
Craig “thought aloud” about possible responses to the new
writing prompts. Using an overhead projector, he read
through the questions with the students, discussed what
was expected, and modeled some appropriate responses.
This modeling included the use of detail in providing explanations and examples, how one might relate two or more
mathematical concepts, and what a real-life application
might be. An additional modeling activity occurred approximately one month later. In addition, Craig and Terry used
the prepared rubric to discuss expectations and student
work throughout the last half of the year. The new prompts
and the explicit guidance on how to respond led to student
feedback that focused more on the need for detail, specific
examples, and mathematical depth.
Discussion
OUR MIDDLE SCHOOL IS LOCATED IN A HIGH SOCIO-
economic area of southwest Washington state with a
largely European-American student population. It uses the
Connected Mathematics Project (CMP) curriculum and
textbook series. Among other things, CMP emphasizes
using student discourse, articulating understanding in
writing, and making connections to previously acquired
knowledge and real-life experiences.
To document any changes as a result of modifying our
approach, an analysis of students’ writing was conducted.
Dramatic improvements were found in student responses
for the entire class after changing the reflective writing
prompts, with the average score rising by approximately
1.5 out of 4 points. Because numeric scores provide only a
small measure of student development, three groups of
two students who received high, average, and low firsttrimester grades, respectively, were randomly selected
for a more thorough analysis of writing. Two writing samples collected between October and January (pre1 and
pre2) that responded to the old prompts and two samples
from March and April (post1 and post2) that responded to
the new prompts were randomly collected from all six of
the students and analyzed using the previously discussed
rubric. The overall scores for these students improved
141 percent, from an average score of 1.4 to 3.3 on the
writing rubric (see table 1). An examination of the length
and specific content of two written responses from both
before and after the prompt change was also made for
these six students. This analysis revealed sharp increases
V OL . 9 , NO. 9 . M A Y 2004
491
Copyright © 2004 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use associated with PBS TeacherLine
only. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.
TABLE 1
Student Scores on Pre- and Postwriting Responses
NAME
PRE1
PRE2
POST1
POST2
Sadie
1
1
2.5
2
Linda
1.5
1
3
3
Sally
1
1
4
4
Teresa
1
1
4
4
Jason
1.5
1.5
3.8
3.5
John
2.5
2.5
2.5
3.5
in the aspects of writing that we had targeted: a greater
focus on connections, a better articulation of understanding algorithms and concepts, and increases in the number
and detail of mathematical applications outside the classroom. Further, the amount of writing produced by the students increased significantly. This was likely the result of
(1) more directed and perhaps more inspiring prompts,
(2) the use of modeling techniques to give guidance to the
students’ writing, and (3) the more detailed written feedback provided by the teachers in conjunction with the
rubric. Two examples are provided to illustrate these
changes.
Linda
Linda is an outgoing, easily distracted student who received a first-trimester grade between a C+ and a D, making her one of two students in the low-grade range whose
writing was analyzed in depth. Her responses to the original writing prompts on January 23 were these:
• To “What? (What have I learned?)” she responded, “I
learned how to put decimals into fractions.”
• To “So what? (What difference does it make?)” she
replied, “Now I know so I can do my work.”
• To “Now what? (What can I do with this information?)”
she noted, “I can use it in my homework.”
Linda’s approach to the writing tasks appears to lack
reflection or any real motivation to do anything but satisfy the requirement. It also appears that her complete
focus is on the mathematical procedures discussed in the
classroom, with no attention to mathematical connections of any kind. On March 26, Linda wrote the following
in response to the revised prompts (spelling errors have
been corrected):
nators the same [student draws an arrow to number 6];
this is the whole and the numerator is the part [student
draws an arrow to number 2]. Then you add 4 + 2 = 6 so
you have 6/6 or 1 whole because 6 is the whole and when
there are 6 of 6 whole you have 1.”
2. “How does your answer to question 1 connect to
other things in mathematics?” “An algorithm relates to
most everything in math. Because you have to explain
everything in math. Because if someone didn’t know how
to do 2 + 2 = 4 then you could use an algorithm to explain.”
3. “How might you use this in the real world?” “I could
use this because I have a little brother who is 2 and he always asks why. So I could use what I know about algorithms to explain why things are that way. Ex. if he asks
me what I was doing homework for I could tell him that it’s
because I want to learn.”
It is immediately clear that the amount of Linda’s writing increased significantly. However, an examination of
the content shows an increase in the quality of the writing,
as well. Linda is much more detailed in her explanations,
and her focus is beyond the description of the procedure.
She is also attempting to find connections to her work with
other concepts and to contexts outside the classroom, as
her comments regarding her understanding of algorithms
show, but she clearly has more progress to make. However, Linda appears to have opened doors for herself in regard to thinking more deeply and powerfully about mathematics. These expanding, but procedurally focused
comments, were common among students in all three
groupings. The expository nature of the writing and the
use of real-life contexts were the most improved areas
across all three student groupings.
Teresa
1. “Explain at least one thing you learned in mathematics this week.” “I learned how to write an algorithm. An algorithm is when you explain the rules to doing a problem.
Ex. 4/6 + 2/6. To do this algorithm you leave the denomi492
Teresa is a quiet and well-liked student who received a
first trimester grade in the B range, making her one of two
students in the average-grade range. As can be seen
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Copyright © 2004 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use associated with PBS TeacherLine
only. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.
below, Teresa underwent similar changes to those of
Linda. On January 2, Teresa wrote the following:
• To “What? (What have I learned?)” she said, “I learned
about decimals.”
• To “So what? (What difference does it make?)” she responded, “It makes a difference because I can write
fractions a different way.”
• To “Now what? (What can I do with this information?)”
she noted, “I can use it when I’m handling money.”
On April 12, Teresa responded in the following way:
1. “Explain at least one thing you learned in mathematics
this week.” “One thing I learned in mathematics this week
was how to multiply fractions. What you do is if the fractions
are mixed numbers you change to improper then you multiply straight across 2 1/3 × 1/2 = 7/3 × 1/2 = 7/6, change
mixed number 3 × 2 = 6, 6 + 1 = 7, multiplying 3 × 2 and 7 × 1.”
2. “How does your answer to question 1 connect to other
things in mathematics?” “This connects to other thing in
math like multiplying whole numbers because you are basically doing the same thing except fractions. 2 × 3 = 6.”
3. “How might you use this in the real world?” “I can see
using this if I want to buy land say that a person had 1/2
of a lot of land and I want to buy 1/2 of that you multiply
1/2 × 1/2 which is 1/4 so I would have 1/4 of a lot.”
The amount of writing, the level of detail, the reflective
nature of the comments, and the ability to think outside
the classroom are all striking differences in these sets of
responses. For example, in response to the second
prompt, Teresa makes an explicit connection to multiplication of whole numbers. Like Linda, who was beginning
to extend her learning outside the classroom, Teresa
makes a specific connection to a measurement context.
Although their writing samples show signs of reflection
and making connections, Linda and Teresa also appear to
be procedurally focused in the content of their writing.
These articulations can help both students
and teachers begin to move beyond this
level of thought.
mentation of, and success in, writing in mathematics are
significant.
We found that it was helpful to our students to convey
clear expectations, demonstrate effective writing in mathematics, and provide clear and specific prompts. Guidelines
for responses and scoring rubrics help in this regard, but
the message must be frequent. Middle school mathematics teachers, in particular, have to assume that their students may not know what complete and coherent mathematics writing looks like. It is important to model the
process early and provide consistent and supportive feedback to student responses. Our students improved their
writing scores dramatically, and they also demonstrated
an ability to make mathematical connections and real-life
applications, albeit at a beginning level, with confidence.
These latter components of writing in mathematics were
almost nonexistent before this study. The writings also
provided individual assessments of levels of understandings that were used to monitor and support the students’
progress.
As teachers gain a better understanding of the questions, prompts, and teaching strategies that yield the greatest learning, the process of writing in mathematics will improve. This scenario will likely lead to a higher level of
understanding of mathematical concepts for most students, which is a coveted goal for teachers everywhere.
References
Goldsby, Dianne S., and Barbara Cozza. “Writing Samples to Understand Mathematical Thinking.” Mathematics Teaching in
the Middle School 7 (May 2002): 517–20.
Miller, L. Diane. “Writing to Learn Mathematics.” Mathematics
Teacher 84 (October 1991): 516–21.
National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.:
NCTM, 2000. Implications for Classroom Practice
ALTHOUGH MANY EDUCATORS AGREE ON
the value of writing in mathematics, how to
elicit quality responses is less clear. Standards-based curricula like CMP promote
writing, but teachers may have difficulty
providing guidance to students in part because of the time demands associated with
this process (Goldsby and Cozza 2002). Any
developments that can enhance the impleV OL . 9 , NO. 9 . M A Y 2004
493
Copyright © 2004 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use associated with PBS TeacherLine
only. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.