Promoting Productive Discourse
EMAT 8990 DISCUSSION PAPER
1
D. WHITE
Promoting Productive Mathematical Classroom
Discourse with Diverse Students
Dorothy Y. White
University of Georgia
Department of Mathematics Education
105 Aderhold Hall
Athens, GA 30602-7124
(706) 542-4096
Fax: (706) 542-4551
Email: dwhite@coe.uga.edu
Running head: PROMOTING PRODUCTIVE MATHEMATICAL DISCOURSE
Submitted for Publication in the Journal of Mathematical Behavior. Do not cite or quote.
The research reported in this material was supported by the National Science Foundation under
Grant number MDR 8954652 and ESI 9454187. The opinions, conclusions, or recommendations
expressed in these materials are those of the author and do not necessarily reflect the views of the
National Science Foundation.
Promoting Productive Discourse
EMAT 8990 DISCUSSION PAPER
D. WHITE
FOR DISCUSSION IN THE EMAT 8990 SEMINAR. DO NOT CITE WITHOUT
PERMISSION OF THE AUTHOR.
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Promoting Productive Discourse
EMAT 8990 DISCUSSION PAPER
Abstract
3
D. WHITE
Productive mathematical classroom discourse allows students to concentrate on sense
making and reasoning; it allows teachers to reflect on students' understanding and to stimulate
mathematical thinking. The focus of the paper is to describe, through classroom vignettes of two
teachers, the importance of including all students in classroom discourse and its influence on
students’ mathematical thinking. Each classroom vignette illustrates one of four themes that
emerged from the classroom discourse: (a) valuing students’ ideas, (b) exploring students’
answers, (c) incorporating students’ background knowledge, and (d) encouraging student-tostudent communication. Recommendations for further research on classroom discourse in
diverse settings are offered.
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Promoting Productive Mathematical Classroom Discourse with Diverse Students
Productive mathematical classroom discourse can facilitate the development of children’s
mathematical thinking (Davis, 1997; Kazemi, 1998; Knuth & Peressini, 2001; Lo & Wheatley,
1994; Martino & Maher, 1999; National Council of Teachers of Mathematics [NCTM], 1996,
1991; Pirie, 1996). Research on classroom discourse often cites the NCTM (1991)
recommendations that mathematics teachers initiate and orchestrate discourse by posing
questions that elicit, engage, and challenge students' thinking; by listening carefully to students'
ideas; and by asking students to clarify and justify their ideas orally and in writing. Classroom
discourse, properly managed, allows the students to concentrate on sense making and reasoning;
it allows teachers to reflect on students' understanding and to stimulate mathematical thinking.
Teachers can stimulate students’ growth of mathematical knowledge by asking more open-ended
questions aimed at problem solving and conceptual understanding (Martino & Maher, 1999).
Productive classroom discourse requires that teachers engage all students in discourse by
monitoring their participation in discussions and deciding when and how to encourage each
student to participate. By actively listening to students' ideas and suggestions, teachers
demonstrate the value they place on each student’s contributions to the thinking of the class.
Thus, if classroom discourse is essential to the learning of mathematics, then researchers and
teachers need to examine the nature and type of communication occurring in classrooms of
diverse student populations. As Hart and Allexsaht-Snider (1996) specifically suggest, we need
more research on teacher development programs that focus explicitly on teachers of diverse
students and the sociocultural contexts of mathematics learning in their school settings.
The purpose of this paper is to describe how two teachers used classroom discourse to
promote the mathematical learning of their diverse students. Through classroom vignettes, I
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EMAT 8990 DISCUSSION PAPER D. WHITE
demonstrate the importance of including all students in classroom discussions and its influence
on students’ mathematical thinking. I begin with a brief overview of the mathematics education
experiences of diverse students, a description of the teachers, their students, and their
pedagogical practices. Next, I present four vignettes to demonstrate how the teachers used
classroom discourse to promote students' mathematical learning. Each vignette illustrates one of
four themes: (a) valuing students’ ideas, (b) exploring students' answers, (c) incorporating
students' background knowledge, and (d) encouraging student-to-student communication. Each
vignette includes a brief explanation of the importance of the type of discourse with respect to
the mathematical content and implications for students’ learning. Finally, I present implications
and recommendations for teacher educators and mathematics education researchers. The
recommendations are designed to help those interested in the educational experiences of diverse
students identify and extend the current research on effective teaching strategies in mathematics
classrooms.
Educational Experiences of Diverse Students
The disparities in mathematics achievement among students are well documented
(Strutchens & Silver, 2000; Tate, 1997). In national mathematics assessments, African American
and Hispanic students continue to score at significantly lower levels than White and Asian
American students. For example, data from the 1996 National Assessment of Educational
Progress ([NAEP], Strutchens & Silver, 2000) found that the average proficiency of African
American and Hispanic students at all grade levels was considerably lower than that of White
students. These differences were especially substantial on tasks that called for extended
responses and complex problem solving. Although African American and Hispanic students have
made achievement gains in recent years, these gains have been on low-level, basic mathematics
skills. As Secada (1992) noted, basic skill proficiency is not enough for “true knowledge and
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EMAT 8990 DISCUSSION PAPER D. WHITE
mastery of mathematics” (p. 630). Instead, all students "need to learn a new set of mathematics
basics that enable them to compute fluently and to solve problems creatively and resourcefully"
(NCTM, 2000a, p. 1).
The poor academic performance of African American and Hispanic students in
mathematics is attributable, in large part, to their educational experiences in mathematics
classrooms (Campbell & Langrall, 1993; Oakes, 1990; Secada, 1992). According to NCTM
(2000b), "students' understanding of mathematics, their ability to use it to solve problems, and
their confidence in, and disposition toward, mathematics are all shaped by the teaching they
encounter in school" (p. 17). Researchers that have examined the educational experiences of
African American and Hispanic students in mathematics report that these students are
disproportionately placed in low-tracked mathematics classes that are largely taught by direct
instruction, rely heavily on worksheets, and cover relatively little of the curriculum (Oakes, 1990;
Secada, 1992). Teachers often believe that a primary goal of instruction is control of minority
students, which can best be achieved in teacher-centered classrooms (Stiff, 1998). In these
classrooms, teachers spend more time directing students on repetitive tasks, remedial work, and
conformity to rules than on developing students’ mathematical competence and autonomous
thinking. However, research supports the view that students do not learn mathematics effectively
when passively listening to teacher directions and disengaged from the learning process. As
Campbell (1998) suggests, "The character of the child is not the issue; the issue is the character
of the instruction" (p. 50).
Descriptions of teachers successfully educating African American and Hispanic students
(Author, 1997, 2000; Gutstein, Lipman, Hernandez & de los Reyes, 1997; Ladson-Billings, 1997;
Malloy, 1997) can help us understand the unique features of improving instruction and learning
for these students. These studies demonstrate that improving the mathematical performance of
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EMAT 8990 DISCUSSION PAPER D. WHITE
African American and Hispanic students requires a classroom climate that promotes their
learning. Malloy (1997) contends that teachers can create a classroom atmosphere that is
conducive to African American students' mathematical learning by: (a) allowing students to be
active in their learning, (b) encouraging high levels of peer interaction, (c) encouraging group
decision making, and (d) avoiding judging any student either verbally or nonverbally on the basis
of the teacher’s biases. Gutstein et al. (1997) propose a three-part model of culturally relevant
mathematics for Mexican American students. The three components are (a) building on students’
informal mathematical knowledge and building on students’ cultural and experiential knowledge,
(b) developing tools of critical mathematical thinking and critical thinking about knowledge in
general, and (c) orientations to students’ culture and experience. In classrooms with these
features, students learn mathematics through a system of instruction that combines the learning
of basic computation skills with higher-order conceptual reasoning. Central to this environment
is the type and nature of the classroom discourse and whether it is accessible to all students.
Methodology
This investigation examined how teachers used classroom discourse to teach mathematics
and whether the discourse enhanced the educational experiences of their diverse student
populations. The research questions were: (1) What was the nature and focus of teachers’
classroom discourse? and (2) How did teachers use classroom discourse to attend to the
mathematical needs of their diverse students? In this section, I present a summary of the teachers’
and students’ characteristics and a description of the teachers’ involvement in Project IMPACT
(Increasing the Mathematical Power of All Children and Teachers). I also provide a description
of the data sources and analysis used in this investigation.
Promoting Productive Discourse
EMAT 8990 DISCUSSION PAPER
Participants
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D. WHITE
The participants were two third-grade teachers and their students in a large, urban school
district located just outside of Washington, DC. The teachers and students were part of a
longitudinal research project entitled Project IMPACT to design, implement, and evaluate a
model for mathematics instruction in schools serving children of diverse ethnic and
socioeconomic backgrounds (for more information on Project IMPACT, see Author, 1997,
2000). The teachers, Ms. Davis and Ms. Tyler, were both White and in their second year of
teaching. Ms. Davis taught 22 students at a language arts magnet school, and Ms. Tyler taught 27
students at a social studies/science magnet school.
The students in Ms. Davis’s and Ms. Tyler’s classes represented various ethnic/racial
groups and were classified into the following categories: Asian, Black, Hispanic, and White. This
racial categorization was based on the school system's policy for classifying students. Asian
students were from Vietnam, Korea, Cambodia, and nations of Southwest Asia, as well as Asian
Americans. Students who were African American, African, Haitian, or from the Caribbean were
considered Black. Hispanic students were students who were Hispanic American, or immigrants
from Central and northern South America, and Spanish-speaking European countries. Any
student who was not considered Asian, Black, or Hispanic, as defined above, was classified as
White. The 22 students in Ms. Davis’s class had the following racial distribution: 18% Asian,
36% Black, 36% Hispanic, and 9% White. Of the 27 students enrolled in Ms. Tyler’s class, 48%
were Black and 52% were White.
Students in both classrooms were evenly mixed across gender but diverse with respect to
their socioeconomic status and mathematical academic performance. More specifically, of the 17
students in Ms. Davis’s where data were available, 12 of the students received free or reducedfee lunch. In Ms. Tyler’s class, 5 of the 23 students where data were available received free or
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EMAT 8990 DISCUSSION PAPER D. WHITE
reduced-fee lunch. Both classrooms were heterogeneous with respect to mathematical
performance. Based on the Project IMPACT 113-item Mid-year Assessment, student scores
ranged from 16-91 in Ms. Davis’s class, and from 32-95 in Ms. Tyler’s class.
Project IMPACT. A major component of Project IMPACT was its teacher enhancement
program, which included a summer inservice program and on-site support during the academic
year. As participants in the project, teachers attended a 22-day grade-specific summer
enhancement inservice program. During this program, project staff addressed issues relating to
(a) adult-level mathematics content; (b) teaching mathematics for understanding, including use of
manipulative materials and integration of mathematical topics; (c) current reform documents and
research on children’s learning of mathematics as well as teaching and learning from a
constructivist perspective; and (d) teaching mathematics in culturally diverse classrooms
including implications of teacher’s expectations, use of praise versus encouragement, grouping
practices. Particular attention was devoted to helping teachers develop techniques for
implementing productive classroom discourse, including the implications of teacher’s questions
and responses for students’ mathematical thinking and participation in class discussions.
The summer inservice program provided time for teachers to experience teaching from a
constructivist perspective, to practice and refine their questioning techniques, and to plan for the
upcoming academic year. For 10 mornings during the inservice, the teachers taught mathematics
to small groups of four to five elementary school children enrolled in summer school who had
either just completed third grade or were entering third grade in the coming school year. Small
debriefing groups with project staff and the other third-grade teachers in their schools followed
morning teaching sessions. For their participation in the summer program, teachers received three
graduate credits and a financial stipend.
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EMAT 8990 DISCUSSION PAPER D. WHITE
During the school year following the Project IMPACT summer program, the teachers
received academic support from an on-site Project IMPACT mathematics specialist assigned to
each school. Throughout the school year, the teachers participated in weekly planning sessions
with the mathematics specialist and the other third-grade teachers at their schools. The
mathematics specialist also assisted the teachers by providing demonstration lessons and helping
in preparing instructional materials.
Data Sources and Analysis
Transcripts of classroom observations, supplemented by my field notes, provided the first
source of data for this study. During the academic year following the Project IMPACT summer
inservice, I observed each teacher teaching mathematics on eight separate occasions from
January to June. In collecting the data, I assumed the passive observer stance in which I sat off to
the side in the front of the room with a pad and tape recorder. Classroom observations were
audiotaped via a remote microphone worn by the teacher that allowed me to record most of her
verbal interactions. In conjunction with the audiotapes, I recorded the teacher’s nonverbal actions
(e.g., writing on the blackboard, distributing materials, observing students, and using
manipulative materials) and her selection of students. After the last classroom observations, I
individually interviewed each teacher to provide data about her perceptions of the classroom
discourse, questioning patterns during mathematics instruction, and whether her views were
consistent with her actual classroom practices. These semi-structured interviews provide the
second source of data for the study.
A separate set of analyses was conducted for each teacher using methods of analytic
induction (Bogdan & Biklen, 1992). I chose a qualitative perspective because it afforded me the
opportunity to describe the teachers’ classroom discourse and questioning patterns in a
Promoting Productive Discourse
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EMAT 8990 DISCUSSION PAPER D. WHITE
naturalistic setting while attending to both the content and context of the discourse (Carlsen,
1991).
Transcripts of classroom observations and field notes were first analyzed by examining
her question and response patterns. These patterns were often a series of questions and responses
rather than a single exchange. For example, when a teacher asked an open-ended question and
selected several students to respond, that exchange was considered one pattern. Four general
questioning patterns emerged, and upon second readings of the transcripts, subsidiary patterns
were identified based on the teachers' responses. These patterns were then categorized into
themes based on the nature and focus of the discourse. The four themes were: (a) valuing
students’ ideas, (b) exploring students' answers, (c) incorporating students' background
knowledge, and (d) encouraging student-to-student communication.
Once the themes were identified and assigned to units of data, these themes were
analyzed to identify the students that were involved in the interactions based on categories across
students' gender and race (Irvine, 1985; Simpson & Erickson, 1983). This analys1s helped
answer the second research question, how did teachers use classroom discourse to attend to the
mathematical needs of their diverse students?
RESULTS
Two Classrooms
Ms. Davis taught mathematics in the morning as the first period of the day. Desks
clustered in groups of four filled her large, well-lit classroom. A crescent-shaped table with
chairs was positioned in one corner of the room for small-group work, and a large carpeted area
in front of the chalkboard was available for students to gather. Each day began with an earlybird mathematics problem for the students to solve as they entered the classroom. Most earlybird problems involved some sort of data collection and representation in which students placed
Promoting Productive Discourse
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EMAT 8990 DISCUSSION PAPER D. WHITE
their answers on either a graph or Venn diagram. Once the class completed the problems, Ms.
Davis gathered the students around the chalkboard to share their answers and solution strategies.
After the early-bird activity, the class discussed the topic for the day and was assigned groups in
which to work. As the children worked on the task, Ms. Davis circulated around the room to
monitor their progress and to ask and answer questions. On some occasions, she would work
with a small group of students while the rest of the class worked individually at their desks.
When time allowed, Ms. Davis followed the small-group work with a whole-class sharing
activity in which students shared their answers and how they solved different problems.
Ms. Tyler taught mathematics in the middle of the day immediately after lunch and
recess. Her small, poorly lit classroom was also arranged with desks clustered in groups of four.
Three corners of the room had a center for a different subject of the curriculum. There was a
reading corner, a science corner, and a mathematics corner. As the lesson began, the class sat at
their desks while they discussed the topic of the day. Whole-class discussions were followed by
students being assigned to work on mathematical problems in groups, pairs, or individually. As
the students worked, Ms. Tyler circulated around the room to monitor the student’s progress and
to ask and answer questions. Whole-class sharing, in which students shared their answers and
solutions strategies, followed the seatwork.
Both Ms. Davis and Ms. Tyler used a hands-on approach to teaching mathematics. They
provided several contexts, such as games and children’s literature books, in addition to
traditional word problems, for students to explore mathematical concepts. They also encouraged
students to work cooperatively and to use manipulative materials to solve problems. In these
classrooms, most mathematics lessons included open-ended problems to allow for multiple
answers and solution paths. Groups were mixed and were constantly rearranged to meet the
needs of the students. As Ms. Tyler explained, “We do a lot more whole group and then break
Promoting Productive Discourse
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down into [smaller] groups. [The groups] are always different because from the whole group you
really see what their strengths and weaknesses are on a given topic.”
During their mathematics lessons, Ms. Tyler and Ms. Davis assumed the role of
facilitator. They led their classes in the problem-solving process but expected the students to
solve problems in their own way. They also expected students to explain their answers and
solution strategies and for the class to judge the mathematical soundness of the answers. The
students were expected to listen to each other and were often reminded of the importance of
listening to each other’s ideas and strategies. The teachers also encouraged students to ask them
or other students questions about a particular answer or strategy. Students could agree or disagree
with answers either verbally or by showing a “thumbs up” or “thumbs down” signal. In Ms.
Davis’s room, there were limited-English-speaking students who would shy away from sharing
their thoughts and ideas with the class. Therefore, Ms. Davis gave all students the option of
sharing with a friend or in writing. According to Ms. Davis, “a lot of times they won’t share
something with the whole group, but they will share it with somebody sitting next to them, or
they can sometimes get ideas from other kids who are sitting next to them.”
What follows are four vignettes to present the type of classroom discourse found in Ms.
Davis’ and Ms. Tyler’s mathematics classes and how these discourse patterns promoted students’
mathematical thinking and learning. The vignettes provide a vivid description of what discourse
looks like in these diverse classrooms and moves beyond dialogues with just two or three
children. As a result, the vignettes are lengthy . In these vignettes, pseudonyms chosen to reflect
the cultural background of the children have been used to mask their identity.
Valuing Students’ Ideas
The beginning of a mathematics lesson sets the tone for what students are expected to do.
Many teachers spend the time telling students the problems they will work on and how they
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EMAT 8990 DISCUSSION PAPER D. WHITE
should be solved. Ms. Davis and Ms. Tyler approached the opening of a lesson as a way to
engage students in the problem-solving process. They began by asking the class what they
noticed about a particular problem, what they thought they were going to work on, or how they
might solve particular problems. Consider the following exchange from the beginning of one of
Ms. Davis's mathematics lessons. Ms. Davis showed the class a poster with some advertised
items (see Figure 1) and asked the class to share what they noticed. The exchange included ten
students (a Black, Hispanic and Asian female, two White females, two Hispanic males, and an
Asian and Black male).
[INSERT FIGURE 1 HERE]
1
T:
Everybody look up here for about 25 seconds and then tell me, be ready
2
to share something that you see up here on my poster. [Pause]. Okay,
3
what kinds of things do you see up here?
[Students correctly state names and prices of coffee, dishwashing liquid and toothpaste.]
4
T:
5
Who sees anything up here that we have not said anything about yet?
Maria?...
6
Maria:
I see a light.
7
T:
You see a light. Okay, that's a light bulb. How much does the light bulb
8
9
cost?
Maria:
Seventy, seven cents.
10 T:
Seventy-seven cents. Could that, could that be dollars? Could that be
11
seventy-seven dollars?
12 Some:
No.
13 T:
Okay, how do you know that it's seventy-seven cents?
Promoting Productive Discourse
EMAT 8990 DISCUSSION PAPER
14 Binh:
Cause it has a c.
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D. WHITE
15 T:
Has the little c with the line through it. Okay, right. There's different
16
ways you can write money, and that's one way you show the cents.
[Another student points out the $77 would be too much to charge for a light bulb. Ms.
Davis agrees, compliments the student on using common sense, and then asks what else
the students see. Twi says she sees the toothpaste.]
17 Twi:
It's not money, cause uhm, it don't have the dollar sign and decimal
18
point.
19 T:
Oh, so if I added--sometimes when you see it in the advertisement, they
20
don't always have it--so if I did that [writes dollar sign], then would that
21
help? Help you to know what it was? Okay, good for you, 'cause I told
22
you that it should have the dollar sign and the decimal point. Is there a
23
way you can tell just by this picture that it's $2.27, though, Twi?
24 Twi:
Yeah.
25 T:
How can you tell that it's $2.27?
26 Twi:
‘Cause if you left, left it like 2, 27, then it would be, then people would
27
think it would cost $227.
28 T:
Okay, but could you tell without what I added up there, that it was
$2.27?
29 Twi:
It would be 227.
30 T:
It would be, all right just by looking up there. Does anybody else have a
31
different way you can tell it was $2.27? .... What do you think the person
32
who made this picture did to show that this was $2.27? ‘Cause you guys
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33
knew it was $2.27. I'm kind of curious to how you knew that.
34
Armando?
35 Armando:
‘Cause the 2 dollars is higher than the cents, so the 2 dollars
36
means it's bigger, and the 27 cents means it's smaller.
37 T:
All right that's how I could tell too, because I saw that this 2 is a bigger,
38
a bigger 2 and then it has the, a little 2 and a little 7 that are sort of raised
39
up. All right good job....
In the first part of the exchange, students began by noticing the names and prices for the
different items. Ms. Davis asked questions to help them recognize different ways money can be
written. She focused on the conventional mathematical symbols, such as the question to Maria
about the cents sign (lines 10-16), and nonconventional ways money is written, as in the
exchange with Twi (lines 17-30) and Armando (lines 30-39). Notice that both Armando and Ms.
Davis focused on the physical size of the numerals rather than the values. As the discussion
progresses, observe how the students’ answers reflect more mathematical ideas.
40 T:
Okay, anything else you see up here? You've already talked about the
41
picture, what else do you see up here? All right Katlin?
42 Katlin:
Uhm, I see uhm, like rows, like you can put something in here.
43 T:
Okay.
44 Katlin:
And also, uhm, if you have to pay $77 dollars for, uhm, the light bulbs,
45
then it would be, you'd probably have a lot of tax on it.
46 T:
A lot of tax? Okay how do you know that?
47 Katlin:
Because 5 cents, it's 5 cents for each dollars, and, uhm, 5 times, 5 times 77
48
is, uhm, that's around 15 or 14 dollars.
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49 T:
How did you know that?
50 Katlin:
I don't know, I just, it's around 9, 10, 12 dollars.
51 T:
Are you estimating or is it, did you figure it out somehow?
52 Katlin:
I figured it out somehow, because, uhm, if you added a whole bunch of 5s,
53
then it's likely to equal dollars, probably more like, it's, it's, well, it's
54
probably more around 14.39.
55 T:
All right so how did you guess that though?
56 Katlin:
Cause, uhm, it has to be, because if you, if you can count the, wait...
57 T:
All right can we come back to that? ‘Cause that's, I'm really curious. All
58
right Katlin said she saw columns or rows up here to put things in.
59
right what about this? What do you see up here? Helen?
60 Helen:
I disagree with that because, if you counted on your fingers like 5, 10, 15,
61
20, 25, 30, 35, 40, 45, 50, and this is only 50 and 10 so, it would be 100 if
62
you have 2, and then...
63 T:
A hundred what?
64 Helen:
A hundred cents, so that would be 1 dollar. And then $2 and then that
65
would be $3, so it's more like, uhm, it's more like, uhm, 3.50.
66 T:
All right that's a really good explanation. That might be something to try
67
out later, okay.
All
In the excerpt above, Ms. Davis wanted all students to know their ideas are valued.
Students of various academic abilities shared their thoughts, and no answer was regarded as
trivial. As a result, the students’ observations varied from noticing different items and their prices
to conceptualizing and computing sales tax. In the case of Katlin, Ms. Davis was not concerned
with the correct amount of tax; instead, she wanted to know how Katlin thought about the
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problem (lines 44-56). The comment by Helen (lines 60-65), showed that the students were
listening to each other and felt comfortable disagreeing with their peers. Asking students to share
what they notice provides the class with information about how people can think about
mathematics problems and tells students that their opinions are valued. Allowing students to
share their thoughts without judgment also increases the number and variety of responses.
Exploring Students' Answers
When Ms. Davis and Ms. Tyler asked questions, their main focus was to explore how
students arrived at their answers. They focused more on students' thinking and their various
solution strategies and less on the correct answer. Both teachers found that by asking students to
explain their answers, they not only learned how the students thought about the problems but also
provided the class with multiple ways to think about and solve problems. Consider the following
example, in which Ms. Davis worked with a group of five students to solve word problems on
division. These students (two Black females, one Hispanic female, one Black male, and one
Hispanic male) had an average achievement score of 26 out of 113, and needed assistance to
solve the assigned class problems. Each child had Unifix cubes and was asked to solve the
problems any way he or she knew how. Ms. Davis used questions to encourage students to share
their answers and solution strategies and to prompt them to correct their errors.
1
Nina:
[Reads problem] "Katlin has a package of cookies to share with her friends.
2
If she wants to give herself and five friends cookies from a package with
3
twenty-four cookies, how many cookies will each person get?"
4
T:
5
6
Okay, what do you have to do? What did you learn in the sentence,
Nina?.... What do you think you have to do? ...
Nina:
How many people.
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7 T:
You have how many people? How many people?
8
Vera:
Six.
9
T:
Six. Why did you say six? How do you know there should be six?... You
10
just guessed? Okay, what told you, what kind of, did something give you a
11
hint that it should be six people?
12 Vera:
Yes.
13 T:
What gave you a hint?
14 Vera:
'Cause it says herself and five friends.
15 T:
Okay, herself and five friends. What do you think "herself and five friends"
16
means? You think that means six? Okay. So if I said, if I said “Katlin and
17
five friends”, how many folks are there? One, two, three, four, five. So if
18
we counted Katlin and all of us that would be six. So you know that there
19
are six people. What else do you know? Ricky, what else do we know
here?
20 Ricky:
There are fourteen cookies.
21 T:
Fourteen cookies? Omar?
22 Omar:
And she wants to give, uhm, twenty-four cookies.
23 T:
All right, we have twenty-four cookies right?
[Ms. Davis asks various students what Katlin wants to do with the cookies, again asks them how
many people there are, and asks what the group should do to solve the problem. She gives them
20 seconds to think about what they should do.]
24 T:
What do you think?
25 Vera:
To, uhm, count out three, and the cookies, uhm, have, uhm, six and put
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26
them here.
20
D. WHITE
27 T:
Put twenty-four in each group of six?... One second Ricky. Got to have a
28
plan before we touch the cubes. So you think we should count out twenty-
29
four cubes and then put twenty-four in each, in six groups?
30 Ricky:
That's what I was thinking.
31 T:
Okay, is that what you think too?
32 Ricky:
I was thinking there are six friends, then we get the twenty-four, get twenty-
33
four. Then you get, uhm, twenty-four cubes of...[Starts to count cubes]
34
Four...
35 T:
Okay, Vera you can start doing what you said you thought you should do.
[Students work individually with Unifix cubes to solve the problem.]
36 Ricky:
Each gets three. Each gets three because there's five more cookies left.
37 T:
Okay, write it down Ricky. Do you want to write down what you're gonna
38
do, Nina?
39 Nina:
I can't think of it.
40 T:
You can't think of it? Okay, why don't you look. Vera and Omar, will you
41
explain to Nina what you're doing? So that she can see what you're doing,
42
and maybe she can get some ideas from what you're doing. Vera, why don't
43
you explain since you said you were finished?
[Vera explains that each friend gets three cookies. She has made a Unifix cube array with 6
columns and 5 rows, using cubes to represent both the cookies and friends. When Ms. Davis
points out 6 cubes standing for friends and 24 standing for cookies, Vera can see that there are 4
rows of cookies]
Promoting Productive Discourse
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EMAT 8990 DISCUSSION PAPER D. WHITE
44 T:
Okay, so how many cookies are there altogether?
45 Vera:
Twenty-four.
46 T:
Twenty-four cookies altogether. Okay, and how many friends are there?
47 Vera:
Six.
48 T:
Six. And how many cookies will each friend get?
49 Vera:
[Counts the cubes in one column.]. Four.
50 T:
Each person will get four? So what would, what would be your number
51
sentence for that?
52 Vera:
Twenty-four divided by..
53 T:
Divided. How many friends did you have?
54 Vera:
Six.
55 T:
Okay, so twenty-four divided by; six friends, equals four cookies each?
56 Vera:
[Nods yes.]
57 T:
Okay write that down. [She turns to Ricky to see what he has done. When
58
he counts his cubes, he realizes he has only 23 cubes and goes back to
59
work.]
Ms. Davis used questioning to help her students think about mathematics problems and to
share their answers. Her main focus at the beginning of the exchange was to have the students
devise a plan before working out the problems, as in lines 1-35. This approach helped Ms. Davis
interpret whether the students understood what the problem was asking and whether they had a
plan for solving it. It also provided students who did not have a problem-solving plan, like Nina
(see lines 38-43), several strategies to use. Ms. Davis asked follow-up questions to ascertain how
the students solved the problem. She also found that by asking questions students corrected their
own mistakes as in the case of Ricky (lines 57-59). Once he heard Vera’s solution, he recounted
Promoting Productive Discourse
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EMAT 8990 DISCUSSION PAPER D. WHITE
and realized that he had twenty-three instead of the needed twenty-four Unifix cubes. This
excerpt also illustrates that Ms. Davis' questioning patterns encouraged students to connect their
problem solving with symbolic representations when she asked her students to write using
number sentences. In reacting to each student’s response, Ms. Davis respected their
mathematical reasoning even when the answer was not correct, as in the case of Ricky (lines 3637).
Incorporating Students' Background Knowledge
Students enter school knowing a lot of mathematics. Their previous school experiences,
coupled with their out-of-school experiences provide them with formal and informal work with
numbers. When students are encouraged to think for themselves, they often incorporate their
knowledge of numbers. Ms. Tyler and Ms. Davis found that while their students’ mathematical
knowledge was still developing, students were always devising new strategies to connect their
formal and informal mathematics experiences. Consider the following example, in which Ms.
Tyler's class worked on solving a word problem about a class trip. The problem asked them how
many more children could fit on the bus if there are 26 people in the class and the bus holds 55
people. One child, Max, had just solved the problem by counting up from twenty-six to fifty-five
by ones. Ms. Tyler asked the other students to share how they solved the problem. This example
included one Black female and two White males.
1
Patrice:
I did the same thing and got twenty-four.
2
T:
Why do you think that happened? Do you think that there's a mistake here or, or
3
do you think maybe you..
4
Patrice:
I think there's a mistake there.
5
T:
Okay.
Promoting Productive Discourse
23
EMAT 8990 DISCUSSION PAPER D. WHITE
6 Patrice:
I started at twenty-six and counted by ones to fifty-five. Well, first I went, I
7
added a ten to it, that's thirty-six, so I'm still not there. Then I added another ten.
8
T:
Okay, thirty-six plus ten is?
9
Patrice:
Forty-six.
10 T:
Okay, so that's two tens [writes 2 tens].
11 Patrice:
And then I, uhm, then I counted by ones and then, see I used the two tens then I
12
counted by ones to see how many ones it would take.
13 T:
From forty-six to fifty-five?
14 Patrice:
Uh huh.
15 T:
Okay, and how many ones between forty-six and fifty-five, if you counted up
16
like forty-seven, forty-eight....
17 Patrice:
Up, I think I did it wrong in my head, ‘cause I got twenty-nine.
18 T:
So did you get twenty-nine when you just did it? ‘Cause you can certainly do it
19
that way too.
20 Patrice:
Yes.
21 T:
So you counted, by tens from twenty-six. You said twenty-six, thirty-six, forty-
22
six, and then at forty-six you counted up by ones, and just now when you did it
23
you realized that it was?
24 Patrice:
Twenty-nine.
25 T:
Twenty-nine.
26 Patrice:
Because if I added another ten, it would be fifty-six and that would be too much.
27 T:
That's a good way of thinking about it. If you added three tens, which is thirty,
28
then that would give you fifty-six, one more than we need. All right, so that's
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EMAT 8990 DISCUSSION PAPER D. WHITE
29
another way of thinking about it. Lucas?
30 Lucas:
Uhm, I did it, I got twenty-nine a different way. I did, uhm, I got, well, I did it, I
31
knew cards was fifty-two, that's cut in half, is twenty-six.
32 T:
Oh.
33 Lucas:
And so and twenty-six, uhm, twenty-six and twenty-six is fifty-two, plus three
34
more is fifty-five, so that's twenty-nine, plus twenty-six. So, uhm, you have
35
twenty-nine more.
36 T:
So you, so you used what you knew about cards, and that there are fifty-two
37
cards in a deck, and half of them, half the deck is twenty-six. So, you knew that
38
it was, that we needed at least twenty-six more, plus what's left over between
39
fifty-two and fifty-five, which is three. You added twenty-six plus three equals
40
twenty-nine. Good thinking, using some background knowledge. Mitch?
41 Mitch:
I subtracted twenty-six from fifty-five.
42 T:
You subtracted twenty-six from fifty-five. Explain what you mean by subtracted.
When students are encouraged to think and solve problems on their own, they use their
background knowledge. That knowledge may be from their schoolwork or from their out-ofschool experiences, as in the solutions proposed by Patrice, Mitch and Lucas. In particular,
Patrice used her knowledge of place value (lines 6-16), Mitch used his experiences with
subtraction (lines 41-42), and Lucas used his experiences with playing cards to solve the problem
(lines 30-40). The exchange with Patrice also highlights the importance of students’ need to feel
comfortable challenging the answers of others (lines 1-5), to explain and justify their answers
(lines 6-28), and to freely make mistakes and learn from those mistakes (lines 15-28).
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EMAT 8990 DISCUSSION PAPER D. WHITE
Encouraging Student-Student Interactions
Ms. Tyler and Ms. Davis wanted their students to judge the correctness of the various
answers and strategies that were offered. They asked questions such as, "How many people
agree or disagree?" and "Why do you agree or disagree?" Both teachers wanted their students to
be responsible for judging the answers and ideas of others; they did not assume the role of
mathematical judge and jury. At the same time, they expected students to explain their answers
and suggestions. Consider the following example, in which the students in Ms. Tyler's class
worked on fact families (Author, 2000). The exchange included twelve students: two Black
males, two Black females, five White males, and two White females. One student had just solved
a word problem, arriving at the number sentence 18 - 8 = 10. In this example, Ms. Tyler wanted
the students to indicate members of a fact family for the equation they wrote.
1
T:
2
Is there another way to move around the numbers, and it'll still go with the
story?... Tammy?
3
Tammy:
Ten minus eight.
4
T:
So if I'm using all--.
5
Tammy:
No, no--.
6
T:
I saw Dennis's hand go up and then come down. I saw Benita's hand go up
7
and then come down. Tell me what you're thinking, Dennis.
8
Dennis:
Uhm, there's, there's one more to do, ten plus eight.
9
T:
Oh, okay. If you want to switch around your, uhm, addition, ten plus eight
10
equals eighteen [writes 10 + 8 = 18]. All right..
[Students suggest other equations, 18 = 10 + 8 and 18 = 8 + 10, which the teacher writes.]
11 T:
Does that work?
Promoting Productive Discourse
EMAT 8990 DISCUSSION PAPER
12 Some:
Yes.
26
D. WHITE
13 T:
Mitch?
14 Mitch:
Eight, uhm, equals eighteen minus ten.
15 T:
Eight equals eighteen minus ten [writes 8 = 18 - 10].
[Students continue to discuss the equation]
16 Sarah:
I disagree with that answer. [points to 8 = 18 – 10.]
17 T:
What?
18 Sarah:
You can't do that. Because you can't take, you can't take eighteen away
19
from ten.
20 T:
Is he taking eighteen away from ten?
21 Sarah:
Uh huh.
22 Some:
No.
23 Sarah:
Yeah, ‘cause it's backwards. He says--. If you flip it over, you can't take ten
24
away from eighteen. I mean you can't take eighteen away from ten. The
25
eighteen and the ten...
26 T:
So you're saying this [writes 8 = 18 - 10] is the same as, this [writes 10-18
27
= 8]?
28 Sarah:
Uh huh.
29 T:
What do you have to say Mitch? Sarah is, Sarah is questioning your, your
30
number sentence. Do you agree with her? Do you disagree? Do you have
31
a reason why you say this does work?
32 Mitch:
The answer is eighteen, but if you flip it over, it doesn't make sense. It's
33
just looks backwards, if the eighteen is really behind the ten.
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EMAT 8990 DISCUSSION PAPER D. WHITE
[The students discuss the meaning of 8 = 18 - 10 and whether it is the same as 8 = 10 - 18.
After some expressions of confusion, Philip tries to clarify the issue.]
34 T:
Philip, can you say what you're saying out loud please?
35 Philip:
Uhm, it, I know what, I think I know what Sarah is trying to say. Uhm, the
36
problem isn't totally turned around, it's just the answer is.
37 T:
... So this [points to 8 = 18 - 10] is the same as this [writes 18 - 10 = 8],
38
only the answer is moved? Is that what you're saying?
39 Philip:
[Nods yes.]
40 T:
Do you agree with that?
41 Sarah:
Yes.
A common misconception regarding the equal sign is that operations on numbers must be
written on the left and answers on the right. In this example, Sarah’s disagreement may be based
on her misconception of the role the equal sign plays in subtraction problems, as in lines 16-24.
Ms. Tyler did not assume the role of authority in verifying students' answers; rather, she posed
the problem back to Mitch to have him explain his answer and say whether he agreed with
Sarah’s comment (lines 29-33). When Mitch was unable to respond and seemed confused, Ms.
Tyler noted Philip's comment and encouraged him to add to the discussion (see line 34). Once
Philip offered his explanation, Ms. Tyler turned back to Sarah to see whether she agreed with
Philip's assertion (lines 35-41). The importance of this exchange is that Ms. Tyler valued the
students’ right to disagree and be confused and that she encouraged others to participate to form a
consensus. She wanted her students to accept or reject answers freely and to talk to one another
before accepting the correct answer.
DISCUSSION
Promoting Productive Discourse
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EMAT 8990 DISCUSSION PAPER D. WHITE
Ms. Davis and Ms. Tyler focused on developing children’s mathematical competence and
autonomous thinking. They encouraged students to solve problems creatively and resourcefully,
thereby developing their students' problem solving abilities and basic computational skills.
Moreover, these teachers’ practices help dispel the myth that African American and Hispanic
students must be told how to think about and solve mathematics problems. To the contrary, these
teachers facilitated mathematical thinking by engaging their students, encouraging them to
critically analyze answers to the questions being posed. That is, the classroom discourse in their
classes was centered on purposeful mathematical talk with genuine student contributions and
interactions (Lo & Wheatley, 1994).
Asking challenging questions and listening to students’ answers and solution strategies
alone are not enough to bring about change in African American and Hispanic students’
mathematical content knowledge. Effective teachers must interpret students’ responses as
indicators of their levels of understanding and adjust their pedagogy accordingly (Author, 2000).
As Martino and Maher (1999) suggest, teachers must become skilled listeners and be prepared to
build on students’ ideas to stimulate further thought. Productive classroom discourse requires
that students' ideas are encouraged, valued, and used to shape instruction. For Ms. Davis and
Ms. Tyler, their instructional practices developed over time. As a result of asking children to
share their thinking every day, children became more fluent and able express their ideas. That
only happens when mathematical discussions become a classroom norm that is negotiated and
changed. Unfortunately, the teaching vignettes presented in this paper concern practices that are
not common in classrooms containing African American and Hispanic students. Too often,
teachers of these students focus on repetitive tasks, remedial work, and conformity to rules,
thereby exerting control over their students. When students are taught in these classrooms, they
are rarely afforded the opportunity to hear and participate in rich mathematical discussions.
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EMAT 8990 DISCUSSION PAPER D. WHITE
African American and Hispanic students, in particular, and all students, in general, need
the types of experiences described in this paper in order to meet the demands of higher
educational standards. They need teachers that orchestrate productive classroom discourse to
shape the learning environment in a positive way, allowing all children to show that they can
learn mathematics and learn from each other. All students need opportunities to solve problems
on their own and share their solutions with peers and teachers. Not until we begin to see as
commonplace the kind of teaching discussed in this paper will African American and Hispanic
students succeed in mathematics. As the vignettes illustrate, students can meet the challenges of
a rich mathematics curriculum when given the opportunity.
CONCLUSION
School systems across the country are engaging in educational reform that demands
higher educational standards and results. Mathematics education has received a large share of
the attention because of the need for schools to produce more quantitatively literate citizens. No
longer can schools allow large number of students, especially African American and Hispanic
students, to leave school underprepared for the technological society in which they will live and
work. According to Strutchens and Silver (2000), the achievement gap will narrow only when
expectations are raised for all students and when students are placed in schools encouraging them
and providing them with means to develop and use their talents and skills.
Our knowledge of the mechanisms of classroom discourse is very rudimentary. We need
much more research on what works and what does not and analyses on why some things work
and some do not. In particular, further research on classroom discourse needs to provide more
evidence of the practices of successful teachers of diverse students. Research must address how
teachers are interacting with students of various racial, economic, and academic backgrounds to
identify areas of strength and those that need improvement. We must critically look at how
Promoting Productive Discourse
30
EMAT 8990 DISCUSSION PAPER D. WHITE
schools function and the implications of grouping and instructional practices on student
achievement. Research on classroom discourse provides an ideal context in which researchers,
educators, and administrators can examine mathematical learning.
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EMAT 8990 DISCUSSION PAPER
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D. WHITE
References
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Knuth, E. & Peressini, D. (2001). Unpacking the nature of discourse in mathematics
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D. WHITE
Buy 3
Pay with $1.00
Pay with $5.00
How much will it be if you buy
3?
Light bulbs
77¢
Toothpaste
227
Coffee
149
Dish liquid
69¢
Figure 1: Poster used by Ms. Davis