THE ROLE OF THE FORMAT OF
ARITHMETICAL TASKS
IN CLASSROOM INTERACTIONS
Adalira Sáenz-Ludlow
Department of
Mathematics
The University of
North Carolina at Charlotte
sae@email.uncc.edu
The analysis of two students’ solutions of
simple addition/subtraction tasks presented in different formats indicates that
the students became more aware of their mental actions and the meaning of the
standard algorithms.
This paper analyzes the mathematical activity and classroom
interactions of two students when they solved addition/subtraction tasks
presented in different formats. The
first section lays out the theoretical rationale for the analysis of the
classroom mathematical activity in terms of the actions and interactions of the
teacher and the students. The second
briefly describes the methodology of a yearlong teaching experiment with fourth
graders. The third presents teaching
episodes that illustrate the mathematical actions and interactions among the
students and the teacher in the context of addition and subtraction tasks.
Theoretical Rationale
One of the goals of the reform movement (NCTM
Professional Standards for Teaching Mathematics, 1991; Principles and Standards
for School Mathematics, 2000) is to promote a shift from a passive
teaching-learning paradigm to an active one. This shift views the
classroom mathematical activity in terms of the students’ and teacher’s
self-awareness of their mathematical actions and interactions. Furthermore, the students' conceptual
development is viewed as the result of a continual and orchestrated collaboration
among the students and the teacher. In
this process, the teacher needs to take into account, among other things, the
students’ current understanding of mathematics, the nature of the instructional
tasks, the nature of the teachers’ questions, the format of the tasks, and what
is expected from the classroom mathematical activity (Cobb, Yackel, and Wood,
1992; Gravemeijer, 1994). When these
elements are coordinated, they sustain a classroom discourse that is based on
understanding and consensual collaboration.
Before we continue, let us clarify the sense in which the
words actions, interactions, and classroom activity are used in this
paper. Actions are considered to be all
acts made by one person that may or may not have the potential of being shared
with others. Interactions are
considered to be all the personal acts that are shared with others. Finally, classroom activity is considered to
comprise all actions and interactions of the participants.
Methodology
Teaching experiment. The teaching experiment consisted of daily
teaching episodes in a fourth grade classroom.
These episodes were characterized by the teacher-student and
student-student discussion of students’ solutions to arithmetical tasks. Tasks were presented verbally or on paper and
in each episode the teacher tried to infer and interpret the students’
mathematical actions so that the interactions among the classroom participants
could be sustained.
Data collection. Daily lessons were videotaped and field notes were kept. In addition to maintaining records of the
mathematical activity of the students, task pages and scrap papers were also
collected.
Format-
and Diagram-Mediated Interactions
In this section we analyze the students’ re-conceptualization
of whole numbers in terms of different units of ten and how this new way to
“see” numbers influenced the generation of mental strategies that provided a
better understanding of the standard algorithms.
At the beginning of the school year the teacher observed that
the students could read three-digit numbers but had no sense of the
relationship between different units of ten.
For example, students could interpret 678 only as 6 hundreds, 7 tens,
and 8 ones but they could not conceptualize it in any other way. Conceptualizing 678, for example, as 67 tens
and 8 ones, or as 6 hundreds and 78 ones was a foreign idea for the
students. Further, it was observed that
the students carried out numerical computations only by following the standard
algorithms in a rote manner.
To provide students with opportunities to develop a better
understanding of the standard algorithms, the teacher began to emphasize mental
computation. To avoid suggesting mental
habits already formed by rote memorization of the algorithms, some of the tasks
purposely displayed the numerals in unconventional ways. For example, in matrices or on the surface
of plane geometric figures.
It was also observed that when students were presented with
paper-and-pencil tasks they were inclined to compute using only the standard
algorithms as computational devices.
But when tasks were presented verbally, students solved them mentally by
using novel strategies for decomposing numbers into different units.
Let us first introduce a classroom episode that indicates
students’ initial adherence to the standard addition algorithm as the only
means for adding numbers. Students were
asked to mentally add 159 and 199. The
task was intended to help students develop mental strategies for adding. After discussing possible ways for rounding
these numbers to the nearest ten and hundred to estimate the sum, the teacher
asked the students to find the sum mentally.
Attempting to give students a hint, the teacher asked, “Could we start
adding the hundreds?” To this question one of the students promptly responded,
“You could but it would screw up the whole problem.” This answer indicates that this student was relying on the
standard algorithm for addition. This
algorithm starts every addition by first adding the digits indicating the units
of one, then adding the digits indicating the units of ten, and so on, in
strict sequential order from right to left.
In the absence of paper and pencil, some students carried out the
algorithm by arranging the numerals vertically in their minds while gesturing
in the air with their fingers as if they were using the standard algorithm on
paper.
Because of daily emphasis on mental computation students
began to decompose numbers into units that served their self-scripted goals to
add and subtract. The diversity of
solutions presented by the students to the class allowed the teacher to
orchestrate discussions about computational strategies. For example, when the teacher asked the
students to mentally add the numbers 759 and 684, Preston, the same child who
weeks before argued that starting with the hundreds “would screw up the whole
problem”, took a different approach. He
explained his mental strategy at the same time that he drew Figure 1 on the
board. The letters Th,
H, T, and O stand for thousands, hundreds, tens, and ones,
respectively. This notation was
collectively selected and commonly used by the students to facilitate the
explanation of mental computations in terms of different units of ten.
Preston: Well,
you take 7 hundreds and 6 hundreds, add them; that would be 13.
Teacher: 13
what?
Preston: 13
hundreds.
Teacher: Okay.
Preston: And
you take 5 hundreds, and 8 hundreds.
Teacher: 5
what and 8 what?
Preston: Oh,
yeah! 5 tens and 8 tens.
Teacher: Okay.
Preston: That would be 13 tens. Then, take 9 tens with 4 tens ... 9 ones
with 4 ones [correcting himself], that would be 9…10-11-12-13. Yeah! 13.
Hey! 13 hundreds, 13 tens, and 13 ones.
Teacher: Magic! 13 of everything! Now that you have 13 of everything, what do you have to do?
Preston: You take 10 hundreds and make 1 thousand
and have 3 hundreds left. You take 10
tens and make 1 hundred and have 3 tens left.
You add 1 hundred to the 3 hundreds to make 4 hundreds. Then, you put together the 3 tens and 1 ten
to make 4 tens. Then, you have left
only 3 ones. So, we have 1 thousand, 4
hundreds, 4 tens, and 3 ones. That is
one thousand four hundred forty-three.
759 684
13
H 13 T 13 O
1
Th 3 H 1 H 3 T 1 T 3
O
1
Th 4 H 4 T 3 O
Figure 1.
Preston’s unit-decomposition strategy to add two numbers
Both the diagram and the dialogue indicate that it was not
difficult for Preston to begin adding first the hundreds and then the other
units of ten in a decreasing order. His
explanation gives us an illustration of the influence of both the
re-conceptualization of numbers into different units and the verbal format of
the tasks on the mathematical activity of the students.
When the teacher posed the addition problem to be solved
mentally, Preston reciprocated her action by devising a mental strategy that
was based on both the decomposition of the two numbers and the use of a
numerical diagram. It is difficult to
infer whether the diagram was the result of Preston’s progressive tinkering
with the numbers while building it up or whether it was simply a representation
of his elaborated mental strategy. The
conceptual understanding manifested in both Preston’s diagram and dialogue with
the teacher indicates a drastic conceptual shift in how he thought about what
he did when he added whole numbers.
It is worth noting that not all the students shifted their ways
of thinking about numbers at the same rate.
Nonetheless, all the students started to generate and imitate novel
solutions like Preston’s and soon they started to use diagrams to generate
results or to explain their strategies.
It is also worth noting that these diagrams, as well as other kinds of
symbolizing, were widely used by the students while interacting with the
teacher and the other students in the class (Sáenz-Ludlow, 1995, 1996, 1998).
The teacher also presented another set of tasks designed to
increase the students' understanding of the mathematical processes that are
built into the computation contained in the standard algorithms for addition
and subtraction. Students were asked to
express in writing the strategies they used.
Figure 2 shows the general format in which these responses were
obtained.
XXX ± YYY
|
||
|
Estimate the answer |
Show a strategy to find the exact answer |
Show another strategy to find the exact answer |
|
|
|
|
|
Verification! Does your answer make sense? |
||
Figure2. General format for addition and subtraction
tasks
The following episode indicates how the teacher’s actions to
modify the format of simple subtraction tasks influenced the students’
actions. Again, the students’ solutions
were mediated by the emergence of numerical diagrams.
Figure 3 shows Hollis’ actions to find the difference between
374 and 92. These actions were
manifested by what he did to symbolize his strategies. The continuous lines are part of Hollis’
solution. The curved dotted lines are
introduced to indicate the direction in which he moved his fingers when
explaining his solution, and the horizontal dotted arrows are introduced to
indicate the direction he followed when operating with the numbers.
Hollis: Let
me show you what I did.
Teacher: Okay.
Hollis: First
I made 300 out of 374, and I made 100 out of 92, and I know that 300 take away
100 is 200.
Teacher: Okay.
Hollis: Here
I took 374 and made 2 hundreds, 17 tens, and 4 ones; and I took 92 as 0
hundreds, 9 tens, and 2 ones. Then, I
did this, 2 hundreds take away 0 hundred is 2 hundreds; 17 tens take away 9
tens is 10-11-12-13-14-15-16-17 [showing eight fingers] 8 tens; 4 ones take away
2 ones is 2 ones.
Teacher: Okay. What else did you do?
Hollis: Oh,
yeah! From 4 ones I took away 2 ones and that is 2. From 7 tens I cannot take away 9 tens, so I go to the 3 and
borrow 1. [He moves his fingers from
one digit to the other.]
Teacher: [interrupting]
go to the 3 what and borrow 1 what?
Hollis: Well 3..., 3 hundreds and borrow 1 hundred, and I know
that 1 hundred is 10 tens, and 10 tens plus 7 tens is 17 tens. 17 tens take
away 9 tens is 8 tens. And 2 hundreds
take away 0 hundreds is 2 hundreds. I
have left two hundred eighty-two.
Teacher: How
do you know that 282 is the correct answer?
Hollis: I added 282 and 92, I learned that last year. You see here 2 ones and 2 ones is 4 ones, 8
tens and 9 tens is 17 tens, 7 tens and carry 1 hundred. 2 hundreds and 1 hundred is 3 hundreds. That is three hundred seventy-four.
|
374 – 92 |
||
|
Estimate the answer |
Show a strategy to find the exact answer |
Show another strategy to find the exact answer |
|
3
0 0 – 1
0 0 2
0 0 |
3 7
4 – 0 9 2 2 H 8 T 2 O |
2 17 3
7 4 – 0 9 2 2
8 2 |
|
Verification! Does your answer make sense? 1 2 8 2 + 0 9 2 3 7 4 |
||
Figure 3.
Hollis’ strategies to subtract two numbers and to verify the result
The format of the task and Hollis’ diagram along with his
gesture-mediated actions indicate several aspects of his thinking. First, in both cases he rounded the numbers
to hundreds but not necessarily to the nearest hundred. Second, he was willing to generate a
strategy by positioning the numbers horizontally and then decomposing the
number 374 into 2 hundreds, 17 tens, and 4 ones. Third, he clearly understood 92 as 0 hundreds, 9 tens, and 2
ones. This decomposition allowed him to
find the subtraction by operating first with the hundred units, then with the
ten units, and finally with the one units at the same time that he generated a
numerical diagram. Fourth, he resorted
to the vertical algorithm for subtraction as another way of subtracting the
numbers, thereby following the convention of “borrowing” and operating with the
digits from right to left. His new
conceptualization of the algorithm was indicated when he explicitly referred to
different units of ten in his explanation.
Fifth, Hollis reconstructed the initial number (374) by adding the
difference between the two given numbers (282) to the number subtracted (092). In so doing, he not only used the standard
addition algorithm but he also explicitly used different units of ten along
with using the convention of “carrying”.
It can be said that Figure 3 indicates Hollis’ modified understanding of
the subtraction and addition algorithms.
In summary, the students’ cognitive actions seemed to depend,
in part, on the pedagogical actions of the teacher to present tasks in
different formats. The teacher’s
actions were product of her ongoing interpretation of students’ re-conceptualizations
of whole numbers. The strategies used
by Preston and Hollis reflected the mental actions that reciprocated the
pedagogical actions of the teacher to present tasks in different formats. The students also generated numerical
diagrams that mediated the communication of their solutions to the class. It is argued here that the interactions
sustained among the students and the teacher fostered the numerical activity of
the students. This argument is supported
by the fact that the students did not show this type of creativity at the beginning
of the school year when they used only the standard algorithms in a rote
manner. What is indicated in the
analysis of the solutions of these two students is a conceptual shift in their
conceptualization of numbers and a re-organization of their understanding of
the standard algorithms for addition and subtraction.
References
Cobb, P.,
Yackel, E., and Wood, T. (1992). Interaction and learning in mathematics
classrooms situations. Educational Studies in Mathematics, 23,
99-122.
Gravemeijer,
K. (1994). Educational development and
developmental research in mathematics education. Journal for Research in Mathematics Education, 5,
443-471.
National
Council of Teachers of Mathematics, Reston (1991). Professional Standards for Teaching Mathematics.
National
Council of Teachers of Mathematics, Reston (2000). Principles and Standards for school Mathematics.
Sáenz-Ludlow, A. (1995).
The emergence of the splitting metaphor in a fourth grade
classroom. In the Proceedings of the
Seventeenth Annual Meeting of the North American Chapter of the International
Group for the Psychology of Mathematics Education, 1, 152-157. Columbus, Ohio: ERIC
Sáenz-Ludlow, A. (1996, October).
The role of children’s numerical diagrams in their conceptualizations
of fractions. Paper
presented at the Annual Meeting of the Semiotic Society of America. Santa Barbara, California.
Sáenz-Ludlow,
A. (1998, April). From addition,
subtraction, and multiplication to the division algorithm: The iconicity of
children’s meaning-making processes.
Paper presented at the Research Presession of the 76th Annual Meeting of
the National Council of Teachers of Mathematics. Washington D. C.
____________________________________
The research reported in this paper was
supported by the National Science Foundation (RED-91557340) and funds provided
by the University of North Carolina at Charlotte. The opinions expressed do not necessarily reflect those of the
Foundation or the University.