Creative Education
2013. Vol.4, No.6, 396-404
Published Online June 2013 in SciRes (
Copyright © 2013 SciR e s . 396
Examining the Types of Mathematical Tasks Used to
Explore the Mathematics Instruction by Elementary
School Teachers
Wei-Min Hsu
Graduate Institute of Mathematics and Science Education, National Pingtung University of Education,
Pingtung, Taiwan
Received April 1st, 2013; revised May 4th, 2013; accepted May 15th, 2013
Copyright © 2013 Wei-Min Hsu. This is an open access article distributed under the Creative Commons Attri-
bution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
This study examined the different types of mathematical tasks used in the classroom to explore the nature
of mathematics instruction of three sixth grade teachers in an elementary school. Case studies, instruc-
tional observations, and classroom artifacts were used to collect data. The results showed that the three
teachers used different types of mathematical tasks and implementation methods. One teacher focused on
high cognitive demand tasks, most of which involved substantial group discussion and students working
cooperatively. Even though the other two also used many high cognitive demand tasks, these were mainly
presented via teacher-student dialogue. By examining the types of mathematical tasks and their imple-
mentation, it was found that the group discussion tasks were generally all high cognitive demand tasks, in
which the students fully explained the solution process. As for the tasks administered through teacher-
student dialogue, due to the usage of large amounts of closed-ended dialogue, the students used low cog-
nition to solve the mathematical tasks and did not have the opportunity to completely explain their think-
ing about the solutions. Thus, in order to fully understand the nature of mathematics instruction by teach-
ers, there should be simultaneous consideration of the types of mathematical tasks used as well as how the
tasks were implemented.
Keywords: Mathematics Instruction; Mathematical Tasks; Task Implementation
Mathematics instruction in the classroom is the key to under-
standing student learning in mathematics (Boaler & Staples,
2008; Gutstein, 2003; Henningsen & Stein, 1997; Mullis, Mar-
tin, Gonzales, & Chrostowski, 2004; Stein, Remillard, & Smith,
2007). However, exactly what does classroom mathematics in-
struction involve? How should one seek to understand and ana-
lyze this instruction in the classroom? Hiebert and Grouws
(2007) believed that instruction is comprised of student-teacher
interactions in the classroom based on the learning of content,
and that the purpose of teacher-student interaction is to achieve
specific learning objectives. Seen from this perspective, there
are two key focuses of classroom mathematics instruction, the
first is the learning content and the second is the interaction be-
tween teachers and students regarding the content. For mathe-
matics, the content to be learned refers to concepts and capa-
bilities associated with numbers and measurement, algebra,
geometry, statistics, and probability (Ministry of Education in
Taiwan, 2003; NCTM, 2000), which are usually presented in
textbooks in the form of mathematical tasks (Henningsen &
Stein, 1997; Stein et al., 2007). At the same time, since the
reform of mathematics instruction in the 1980s, teachers have
emphasized allowing students to learn and understand mathe-
matical concepts via the mathematical tasks they provide, tasks
whereby students have opportunities to apply their mathematic-
cal abilities and techniques (Cowan, 2006). In addition, re-
search has shown that the mathematics instruction provided by
teachers generally focuses on students learning from the mathe-
matical tasks in the textbook (Grouws, Smith, & Sztajn, 2004).
Thus, it is clear that mathematics instruction in the classroom
focuses on mathematical tasks, and the implementation of these
tasks typically involves teacher-student interaction in order to
facilitate mathematics learning. Hence, the implementation of
mathematical tasks in the classroom is a productive way to ex-
plore the mathematics instruction of teachers.
A number of studies have focused on using mathematical
tasks to explore and analyze the instructional performance of
teachers. Henningsen and Stein (1997), for example, used the
types of and the implementation of mathematical tasks to ex-
plore their influence on teaching methods and their results on
stu den t m ath ematics learning. Artzt and Armour- Thomas (2002)
used the mathematical tasks provided, the learning environment
created, and the student-teacher interaction to analyze teacher
performance in mathematics instruction. Silver, Mesa, Morris,
Star and Benken (2009) used the types of mathematical tasks
and the instructional characteristics adopted by the teachers in
the classroom to understand the relationship between the ma-
thematics instruction of the teachers and the students’ learning.
These studies all productively used the types of and the
implementation of mathematical tasks by teachers in the class-
room to explore the instructional performance of the teachers.
The types of mathematical tasks can be categorized into low
cognitive demand and high cognitive demand, based on the
cognitive demands required of the students when solving the
tasks, and it has been shown that these different tasks produce
different implementation methods (Stein, Smith, Henningsen,
& Silver, 2000). After several instances of mathematics cur-
ricular reform, middle and elementary schools in Taiwan have
moved from emphasizing knowledge acquisition and knowl-
edge construction to the cultivation of abilities (Chung, 2005);
at the same time instruction has turned from the previous em-
phasis on mathematical knowledge structure, basic computa-
tional techniques and practicing routine tasks, to emphasizing
the active construction of mathematical knowledge, thus creat-
ing a balance between conceptual understanding and adept
calculations. These changes have not only affected the presen-
tation of mathematical tasks in textbooks but have also affected
the implementation of these mathematical tasks in the class-
rooms. However, research has found that when teachers face
mathematics instruction reform, they often show a resistance to
changing their instruction (Rodriguez, 2005). Teacher instruc-
tion is still fundamentally based on traditional lectures that
utilize low cognitive demand mathematical tasks (Boaler &
Brodie, 2004; Weiss, Banilower, McMahon, & Smith, 2001).
Thus, the type of and the implementation of mathematical tasks
by teachers after nearly 20 years of mathematical curriculum
and instruction reform in Taiwan are worthy of study. This
study would like to combine with the types of and implementa-
tion methods of mathematical tasks to develop a rich descrip-
tion and deep understanding of mathematics instruction in the
classroom, which can in turn be used for further explorations of
mathematics classroom instruction in Taiwan.
Literature Review
Trends and Related Observations in Mathematics
In the first half of the 20th century, mathematics instruction
in Taiwan generally followed the three rules of learning pro-
posed by Thorndike, which emphasized the connection between
stimulation and response. Subsequently, the perspective of con-
structivism in learning has become widely accepted. In this
perspective students’ preexisting knowledge and experience are
emphasized, as well as the importance of active learning in or-
der to achieve mathematics learning with understanding (Wil-
loughby, 2000). In general terms, the trend of mathematics
reform in various countries has changed the perspective on the
learner to that of an individual actively constructing mathe-
matical knowledge (Becker & Selter, 1996; NCTM, 2000). At
the same time, in a high-tech and rapidly changing society, stu-
dents are no longer merely required to have basic techniques
but also need to develop abilities in areas such as diverse view-
points, leadership, interpersonal relationships, team work and
adaptability (Lott & Souhrada, 2000). Thus, mathematics in-
struction at school not only covers mathematical concepts and
computational abilities but also student communication, prob-
lem-solving, deduction, and evaluation (Ministry of Education,
2003; NCTM, 2000), in order to attend to the future needs of
life and society.
In summary, changes in the perspectives on the learning
process have caused mathematics instruction to focus on stu-
dent learning emphasizing student-centered approaches in order
to promote student mathematics comprehension (Kilpatrick &
Silver, 2000; NCTM, 2000). Classroom activities in mathemat-
ics instruction are no longer merely based on an information
transmission model; they now seek to create an environment
that stimulates student thinking, provides learning and problem-
solving activities, and provides guidance and support for prob-
lem-solving (Verschaffel & De Corte, 1996). Feldman (2003)
pointed out that mathematics instruction should provide suit-
able mathematical challenges, as well as assistance and strate-
gies during the learning process to help student learning. Fuson
et al. (2000) further proposed that mathematics instruction
should begin with what is meaningful to the students, in order
to deve lop a culture b ased on comprehension and mutual assis-
tance in cooperative mathematics dialogues. Anderson (2003)
added that the focus of mathematics instruction should be on
student thinking, and that the development of fair learning com-
munities to achieve mathematics instruction with comprehen-
sion should be the target. Seen from these trends and observa-
tions of mathematics instruction, it is possible to understand
that the current types of and methods for administering mathe-
matical tasks in the classroom should be based on providing
meaningful and suitably challenging mathematical tasks to stu-
dents in a cooperative dialogue environment.
Types of Tasks and Methods of Task Implementation
Mathematical tasks include the problems and practice activi-
ties used by teachers in the mathematics classroom. A mathe-
matical task will take up a certain work period in the classroom,
during which students are expected to work hard at learning
specific mathematical concepts (Henningsen & Stein, 1997).
Based on this definition, mathematical tasks may be simple
equations, word problems, or measurement activities. The cog-
nitive demand required of the students when solving tasks can
be classified into four levels: memorization, procedures without
connections, procedures with connections, and doing mathe-
matics (Stein et al., 2000). For mathematical tasks that involve
memorization or procedures without connections, the students
only need to employ rote memorization or mechanical use of
rules to successfully solve the tasks. They do not need real
comprehension of concepts involved. These two types of tasks
are considered low cognitive demand tasks, because they do not
much cognitive demand on the students. As for tasks involving
procedures with connections or doing mathematics, these are
high cognitive demand tasks that require real comprehension of
concepts and ideas as well as the selection of suitable strategies,
including engaging in deduction, induction, and proof, in order
to solve the tasks. For example, low cognitive demand tasks
often focus on recall of basic facts and proficiency in computa-
tional techniques, whereas high cognitive demand tasks empha-
size solving, deducing, and applying. Examples of the four
types of different mathematical tasks are shown in Table 1.
Different types of mathematical tasks tend to lead to differ-
ences in teacher-student interaction and in how the tasks are
presented in the classroom. Low cognitive demand mathemati-
cal tasks emphasize using memorized formulas or relationships
to solve problems and require proficiency in computational
procedures; thus, unidirectional or closed methods are the norm
for teacher-student interaction. High cognitive demand mathe-
matical tasks involve more complex information and mental
processing and thus require students to create problem-solving
Copyright © 2013 SciRe s . 397
Table 1.
Examples of the four types of mathematical tasks.
Task type Example and possible response from students
Memorization What are the fractions and perc entages equal to
1/2? (Write out the answers im mediately based
on the definition)
Procedures without
Convert 3/8 to a fraction and a perce n tage.
(Find the answer through computational
Procedures with
Use the hundred cell board to mark f ractions and
percentag es equal to 3 / 5. (Need to connect
Doing math e m atics
In a 4 × 10 cell board, shade in 6 cells, a n d then
explain how to determine the pe rcentage and
fraction of the shaded surface. (Need to select
suitable m ethods to solve and explain)
Note: Modified from Implementing standards-based mathematics instruction: A
casebook for pro fess ional develo pmen t (p. 13), by Stein et al., 2000, NY: Teacher
College. The content in parentheses shows possible s tudent responses.
methods and use a deep understanding of mathematical con-
cepts during the process of thinking, discussing, and exploring
(Silver et al., 2009). Thus, open and bidirectional interactive
methods can be used to provide students with more opportuni-
ties to explore, interpret, and explain the concepts. For low cog-
nitive demand mathematics tasks, implementation is often based
on teacher lecture, and the learning focus is generally on de-
veloping proficiency of problem-solving techniques. The pres-
entation of high cognitive demand tasks is primarily based on
discussion and cooperative problem-solving. In addition to un-
derstanding concepts, the learning focus is also on the cultiva-
tion of abilities in thinking, communication, and logic (Stein et
al., 2007). This shows that the ty pe of mathematical task not only
affects the learning focus but also the instructional implementation.
However, the type of mathematical task cannot be used to
singularly determine the implementation methods used in the
classroom, because the presentation methods may differ from
the original demands of the mathematical task type. Hen-
ningsen and Stein (1997) found that mathematical tasks that
were originally high cognition tasks might be simplified due to
an improper time distribution, a lack of suitable learning moti-
vation, a lack of the students’ preexisting knowledge, or class
management issues. Boaler and Brodie (2004) found that in
both traditional and revised mathematics textbooks, teachers
generally focused on lecturing about rules and procedures. The
TIMSS 1999’s instructional video research showed that al-
though 17% of the mathematical tasks selected by teachers for
use in instruction were high cognitive demand tasks, they were
implemented as procedural practices (Stein et al., 2007). Thus,
the exploration and understanding of mathematics instruction
teachers requires not only identifying and understanding the
types of mathematical tasks used in the classroom, but also
examining the implementation of the mathematical tasks.
Previous research or literature has pointed out that the im-
plementation of mathematical tasks by teachers in the class-
room is a complex and highly personalized process (Hen-
ningsen & Stein, 1997; Stein et al., 2007). At the same time,
few studies have focused on the classroom implementation of
mathematical tasks to analyze and explore teacher mathematics
instruction (such as Silver et al., 2009). Highly personalized
complex processes and research issues for initial exploration
should be examined using case studies. This research presents
case studies on three sixth-grade teachers with different back-
grounds. Video recordings were made and documents were col-
lected during the course of one semester, in order to understand
the types of and implementation methods of the mathematical
tasks used by the teachers in the classroom. These two data
sources were analyzed to understand the mathematics instruc-
tion demonstrated by the case teachers. Past assertions in the
literature about trends in mathematics instruction, including the
classifications of the types of mathematical tasks, the imple-
mentation methods and their relationships, were all used as
references for data collection and analysis.
This study selected three sixth-grade teachers with different
academic and teaching experience, as well as different teaching
locations, as the case study subjects. Pseudonyms were given to
the teachers to protect their identity. Lian had 15 years of in-
structional experience, had majored in education at university,
was teaching at an urban school, and had worked as a mathe-
matics instruction counselor for one year. Mike had six years of
instructional experience, had majored in mathematics at univer-
sity, was teaching at an urban school, and had five years of
teaching experience at a mathematics cram school. Yan had
nine years of instructional experience, had majored in mathe-
matics at university, was teaching at a rural school, and had
worked as an information instruction counselor for three years.
In the past, Lian’s mathematics instruction had mixed group
discussions and teacher lectures, depending upon student learn-
ing performance. Mike and Yan primarily used lectures as their
instructional method. The three teachers all stated that their
mathematics instruction considered course progress, the results
of student evaluations, and school policy.
Data Collection
The period of data collection in this study was from February
2009 to June 2009. Instructional observation videos and docu-
ment records were primarily used to collect data, which was
compared with definitions in the literature in order to under-
stand the mathematics tasks and methods administered by the
teachers. At the same time, the possible relationships between
the tasks and the implementation methods of mathematics tasks
were considered. In the following, the methods and focuses of
the different data collection methods are explained, as well as
their relationship with the research purpose.
In order to understand how the three teachers presented
mathematical tasks in their classrooms, instructional observa-
tions and video recordings were used to collect data. One class
was recorded each week for each teacher during the chosen
semester. Not counting the three examination weeks, and the
early conclusion of the course for the sixth grade, each case
teacher was observed and video recorded 12 or 13 times. The
observed instructional units included 7 - 8 units about numbers
and measurement, algebra, geometry and statistics. The ob-
served content was not the same for each unit. Sometimes the
first class for a unit was observed, and sometimes it was a mid-
dle class or the last class of a unit. In other words, the content
Copyright © 2013 SciRe s .
observed for the three teachers covered different mathematical
topics and was observed in different orders, in order to repre-
sent the overall implementation of the mathematical tasks by
the teachers. The focus of the instructional observations fol-
lowed the suggestions by Stein et al. (2007) and Silver et al.
(2009), where in addition to coding the types of mathematical
tasks used by the teachers in the classroom, the activities used
when presenting the mathematical tasks and the methods of
teacher-student interaction were also recorded in order to un-
derstand the purposes of the teachers’ implementation of the
mathematical tasks.
Document collection refers to collecting the problem-solving
records left by student s when they participate in proble m-solving
activities in the classroom. Such paper-pencil records include
group or individual problem-solving records (including equa-
tions and diagrams), which are then used to compare with and
calibrate with other collected data, to understand the focus of
the types and implementation of mathematical tasks by teachers.
In addition, after each teacher completed their teaching,
semi-structured interviews were immediately conducted fo-
cused on the sources of the mathematical tasks used and on the
implementation methods employed. Questions such as “Why
do you choose this type of task today?” and “Do y ou usually let
students discuss different problem-solving methods?” were
used to understand the thinking process of the case teachers in
presenting mathematics tasks.
Data Processing and An a l ysis
The data collected in this study were primarily instructional
observation videos and document records, which were sup-
ported by teacher interview data. Each type of data was con-
verted to a text or image for encoding. Encoding was carried
out through the principle of person-date-type. Lian, Mike and
Yan were denoted by the letters L, M and Y, and the students
were represented as S. Dates were encoded as month-date using
four digits. The data types were represented using the first two
letters of the English word (such as observation = Ob; docu-
ment = Do), to serve as the basis for future data presentation.
As part of analyzing the observation data, counting and sum-
marization of the mathematical tasks was conducted. In count-
ing the mathematical tasks, even though there were different
methods of presenting the mathematical tasks (group discus-
sions or teacher-student dialogue), all the case teachers first
presented a mathematical task (word problems, diagram prob-
lems or simple calculations), and then asked the students to
engage in either group discussions, individual problem-solving,
or teacher-student dialogue. Thus, one mathematical task was
counted as the task presented by the teacher, the interaction
based on the task, and the student problem-solving records or
answer content. If the teacher repeatedly used dialogue to guide
the students in problem solving or in comprehending the same
task, but the explored or learned mathematical concept was the
same, it was still counted as one task. The following dialogue
provides an example:
Y: There is a red banner at the front of the store, which
says “25% off discount coupon,” so Lizhi bought herself a
2400 dollar dress. How much worth of coupon should
Lizhi pay? Compare it first, what is the discount?
SS: 75% off.
Y: Is it 75% off? …Is it? Are you sure? Then it would be
very cheap.
SS: Yes (some people shook their heads).
Y: S1 is shaking your head, why? What is wrong?
S1: It’s not the same as in the front… (dialogue was used
to clarify the meaning of “off” in the question, and then
the students were asked to individually solve the problem,
In terms of coding the mathematical tasks, the classification
system developed by Stein et al. (2000) was used to divide the
tasks into high and low cognitive demand groups, as shown in
Table 1. The data used for coding the presenting of mathe-
matical tasks included the teacher-student interaction and class
activities associated with the mathematical tasks. As part of the
counting and classification of the mathematical tasks, the re-
searchers first clarified the definition of a task and discussed the
classification standards, and then individually carried out clas-
sification and accounting based on transcripts of the recorded
class periods of the three case teachers. Next, the researchers
discussed and modified any initial (and different) counting and
classification results. The initial results showed that the count-
ing of mathematical tasks was identical, except for Mike. The
reason for the difference was that when Mike was guiding stu-
dents to understand the formulas for volume, he generally used
cuboids as examples, and then introduced formulas for trape-
zoids and triangular prisms, hoping to arrive at the conclusion
of cube volume being base surface × height (M0227Ob). This
was originally counted as the trapezoid volume, but after dis-
cussion and a comparison of the definition of mathematical
tasks, it was thought that this should be counted as one task,
because the purpose was to lead to a general principle for the
formulas for volume. In terms of classifying the mathematical
tasks, there were different interpretations of the requirements
for the mathematical tasks used by teachers. For instance, when
Mike was leading students toward the volume formulas, one
analyst believed that the item emphasized memorization in
problem-solving. The other believed that the whole purpose
was to lead to common volume formulas, but that it was being
don e th ro ug h la rge am ou nts of cl o sed -ended dia lo gue (M0227Ob),
so the task itself was still a high cognitive task with connections.
After the two analysts reached consensus, they analyzed an-
other class by Mike with greater differences; finally reaching
the same conclusion for the counting and classifications of
mathematical tasks without further analysis.
The strategy of continued comparison as well as engaging in
calibration, categorization, and statistical analysis was used in
order to summarize data and theories and to develop a deeper
understanding of the types and methods of mathematics tasks
administered by teachers.
Types of Mathematical Tasks Used by the Case
Mathematical tasks are classified according to the cognitive
demands required when solving them. Formula memorization,
definitions, and one-step unit conversions are classified as the
memory type, and direct measurement and reading, equation
computations (applying known fixed procedures), and single-
step word problems (tasks lacking a challenging character) are
classified as having a lack of connection; these two types are
both low cognitive demand tasks. Two-step word problems
(requiring choices and decisions), open observations and com-
Copyright © 2013 SciRe s . 399
parisons (requiring comparison and evaluation), and tasks em-
phasizing the connection between representations are classified
as procedures with connections. Doing mathematics requires
the students to be liberated from complex context and to have
the ability to use suitable strategies and representations; these
two types of tasks are both high cognitive demand tasks. In one
semester, during 13 periods of observation, Lian administered a
total of 37 mathematical tasks, of which 10 were low cognitive
demand and 27 were high cognitive demand. Mike adminis-
tered 32 mathematical tasks, of which 11 were low cognitive
demand and 21 were high cognitive demand, and Yan adminis-
tered 43 mathematical tasks, of which 23 were low cognitive
demand and 20 were high cognitive demand. The statistics and
different types of mathematical tasks used by the three case
teachers are shown in Table 2.
As shown in Table 2 the types of mathematical tasks pre-
sented by Lian and Mike focused on high cognitive demand
tasks, while Yan focused on low cognitive demand tasks. Nei-
ther Mike nor Yan implemented “doing mathematics” types of
tasks. Regarding memorization tasks, the three teachers gener-
ally presented tasks that required formulas (such as applying
volume formulas to find the volume of cubes, M0227Ob) and
using definitions to solve problems (such as the definition of
the median, L0318Ob, finding the definition of speed, Y0415
Ob). Tasks presented that lacked connections were generally
the application of fixed procedures in problem-solving tasks,
such as solving equations and reading statistical charts, both of
which emphasize using known and fixed procedures to solve
problems (such as reading the position of an object using Car- -
tesian coordinates, L0430Ob, reading the percentage of books
borrowed by different classes, M0424Ob) with fewer single-
step word problems that have lower task complexity (such as
the statistical amount and pie chart unit, L0312).
Tasks with connections were the ones most frequently im-
plemented by the three teachers. These tasks included open ob-
servations and class discussions (such as providing two-di-
mensional or three-dimensional images so that students can
observe regularities, similarities and differences, M0410Ob and
Y0408Ob), two-step word problems, and tasks requiring mean-
ingful connections (such as asking for the connection between
composite images and equations, L0319Ob). For instance, Yan
used pictorial images of building blocks (shown on an elec-
tronic whiteboard) so that the students could think about possi-
ble solutions. Yan then combined the students’ equations and
the block images, to verify the meanings of the equations writ-
ten by the students. At the same time, comparisons of the dif-
ferent equations and images were used so that the students
could understand the nature and meaning of the distributive
property of multiplication. During the activity, the following
Table 2.
Types of mathematical tasks used by the three case teachers and their
presentation methods.
Task type Lian Mike Yan
Memorization 1 2 8
Procedures without connections 9 9 15
Procedures with conne ctions 25 21 20
Doing mathematics 2 0 0
Total 37 32 43
dialogue was recorded:
Y: There are two piles of blocks, A and B. There are 12
blocks in A and 8 in B. Now, there are two types of
blocks in the box, 5 piles of A and 5 piles of B. So I ask
you, how many blocks are in the box? (Students carry out
individual practice, and some are invited to come to the
Y: S1, please explain these equations. What are you cal-
culating in the first equation? (S1’s equation is 12 × 5 =
60, 8 × 5 = 40, 60 × 40 = 100, S10603Do).
S1: There are 5 piles of 12 blocks, so 12 × 5…
Y: OK, in his first equation, there are 12 blocks in the A
pile, and there are 5 piles, so all of A is 60 blocks. Is that
what you mean?
S1: (Nods, Yan and S1 verify the meaning of each symbol
represented in the equation)…
Y: Is it possible to combine these three equations into one
(S1 shakes head)? S2 please tell us why 12 + 8? (S2’s
equation is (12 + 8) × 5 = 100, S20603Do).
S2: Add the two piles.
Y: Which two piles do you mean?
S2: A and B.
Y: one A plus one B, right?
S2: (Nods, Yan and S1 verify the meaning of what each
equation means).
Y: Is there anyone who can explain it more clearly for
him? Actually S2 knows what it means, but he doesn’t
know how to explain it …
S3: He first calculated the numbers in piles A and B, then,
as there are five piles each, multiplied by 5.
Y: Can I take this, and take this, and count A and B as one
pile… Do you know what I mean? I will show you (opens
the electronic white board image, on the top right, and
uses icons to explain the equation meanings to students.
Irrelevant dialogue is discarded, Y0603Ob).
From examining the paper-pencil solution records of the
students, it was possible to better understand the types of
mathematical tasks implemented by the teachers. For instance,
Lian’s implementation of mathematical tasks frequently re-
quired the students to write what each equation meant, as well
as the connection to the image. For instance, in the composite
image surfaces, the students not only showed the connections
between the equations and images, there were even groups that
thought of repeating the original composite images; after the
rectangle surface was subtracted from the semi-circle surface,
they could obtain the sum of the two shaded areas (see Figure
1). These thought processes during problem solving were
Figure 1.
The paper-pencil solution records of the students.
Copyright © 2013 SciRe s .
praised by the other students (L0319Ob).
Only Lian implemented “doing mathematics” tasks, and
these appeared in the commutative property unit about multi-
plication. One of the problems required the students to substi-
tute numbers into equations in order to illustrate the commuta-
tive property in multiplication. Afterwards, Lian further asked
the students to list word problems that corresponded to a × b +
c × b or (a + c) × b:
L: who can make a similar word problem? (pointing to the
two equations a × b + c × b and (a + c) × b on the black-
S1: Older brother went to the stationery store to buy b
pencils for a dollars each, and also bought b ballpoint
pens for c dollars, how much did he need to pay the store-
L: If you use this, what does it mean? (pointing to a × b +
c × b on the blackboard).
SS: the two are used to calculate how much each is, and
then add them.
L: Which two? How much is each?
SS: Money, the money to pay…
L: If so, what does it mea n? S2, tell me what this me ans?
(pointing to (a + c) × b on the blackboard).
S2: Add up the prices first, and then multiply it by that…
L: The price of which?
S2: Add the prices of the pencils and ballpoint pens.
L: Right, this is the price of one ballpoint pen and one
pencil each… this is the price of a ballpoint pen and a
pencil (pointing to (a + c) on the blackboard), this is the
sum. So the total you have bought is … (Irrelevant dia-
logue is discarded, L0521Ob).
The students needed to first understand the meaning of the
equations before they could write the corresponding word pro-
blems, so this was classified as “doing mathematics.” During
the dialogue, Lian confirmed the textual meanings correspond-
ing to the equations and the contextual meanings corresponding
to a × b + c × b and (a + c) × b, and then used the textual
meanings to let the students understand the distributive prop-
erty in multiplication. The other question required the students
to write word problems corresponding to “a ÷ b + a ÷ c and a ÷
(b + c),” and then see whether the two equations were the same
The Implementation of Mathematical Tasks by the
Case Teachers
Even though all three teachers used the implementation meth-
ods of group discussion, teacher-student dialogue, and individ-
ual practice, Lian’s implementation of mathematical tasks was
primarily based on group discussion (29/37). Mike’s relied on
mixed group discussion and teacher-student dialogue (8/32 and
17/32, respectively), and Yan focused on teacher-student dia-
logue (32/43). Seen from Table 3, Lian was inclined toward
using cooperative learning to implement mathematical tasks,
while Mike and Yan were inclined toward using teacher-student
dialogue and direct presentation in their implementation.
Further analysis of the task implementation and types of
mathematical tasks data showed that the mathematical tasks
presented through group discussion were almost all high cogni-
tive demand tasks, although the teachers differed on teacher-
student dialogue methods and the implementation of individual
Table 3.
Statistical summary of the teachers’ implementation of mathematical
Implementation of math e matical tasks Lian Mike Yan
Group disc us sion 29 8 3
Teacher-st udent dialogue 5 17 32
Individual practice 3 7 8
Total 37 32 43
practice. However, overall there were more low cognitive de-
mand tasks than high cognitive demand tasks. Lian imple-
mented 29 mathematical tasks using group discussion, of which
26 were high cognitive demand tasks and three were low cogni-
tive demand tasks, such as asking the students to observe the
changes in length and width of two photographs in the scale
unit, which were low cognitive demand tasks using solutions
created by known procedures (L0423Ob). In eight group dis-
cussion tasks, Mike implemented seven high cognitive demand
tasks, of which one covered statistical chart manufacturing and
computation (M0424Ob), which had low cognitive demand, as
known procedures were used to solve single-step tasks. Yan
implemented three group discussion tasks, all of which were
high cognitive demand tasks.
Among the mathematical tasks implemented using teacher-
student dialogue, Lian administered four out of five low cogni-
tive demand tasks, such as explanations on median and mode
(L0312Ob). The coordinates unit used the dialogue method to
request students to point out the coordinates of students and
locations in the image (L0430Ob). Of Mike’s 17 tasks, seven
were low cognitive demand and ten are high cognitive demand,
but only one allowed the students to engage in open observa-
tions and comparisons (shrunken images and scales unit,
M0410Ob); the remaining nine were highly structured and
guided by the teacher (cutting the task into several small tasks).
Of Yan’s 32 tasks, 20 were low cognitive demand and 12 were
high cognitive demand, but ten were conducted with closed
dialogue under highly structured guidance, and only two en-
gaged in dialogue with open questions or emphasized connec-
tions, such as: “Hongjie says it took me 120 seconds to skate
two laps, but Ziyi says it only took me 2.5 minutes to skate two
laps. If you were the coach, who would you choose to be the
skater” (Y0415Ob).
Regarding the tasks implemented through individual practice,
Lian’s three tasks were all low cognitive demand tasks. In
Mike’s seven tasks, five tasks were low cognitive demand and
the other two were two-step word problems from the textbook
(M0525Ob). Yan was unique, in that only two out of the eight
individual practice tasks were low cognitive demand. The rest
were open operations, observations, and comparisons (such as
finding the center of a circle, Y0318O and the elements and
rules for the formation of rectangles, Y0408Ob), or were two-
step complex high cognitive demand tasks.
Teacher-Student Interaction in Implementing
Mathematical Tasks
The methods of teacher-student interaction were affected by
the methods of implementing the mathematical tasks. Those
implemented through group discussion was open dialogue,
which means the teachers asked open questions, the students
Copyright © 2013 SciRe s . 401
could use the methods they understood to solve the tasks, as
well as explain the complete thought processes and results of
the solution. If the tasks were implemented using teacher-stu-
dent dialogue, then it was primarily closed dialogue, which
means the questions asked were closed with a fixed answer. In
these situations the students used short answers rather than
explaining their complete thought process for the solution. Of
the 37 mathematical tasks, Lian had three individual practice
tasks without any teacher-student interaction (the students prac-
ticed until the end of class, L0312Ob), and two group discus-
sion tasks that were discussed until the end of class (L0226Ob,
L0430Ob) and could not be counted. There were 27 tasks were
group discussion mathematical tasks, all of which used open
dialogue, asking each group to use their own methods and un-
derstanding to solve the tasks, and then providing the students
with opportunities to completely solve the tasks as well as op-
portunities to ask questions. The other 5 tasks carried out thr-
ough teacher-student dialogue, only one task asked the students
to write word problems corresponding to the equation (a + c) ×
b and then engage in open dialogue (L0521Ob), the other four
carried out closed dialogue. Of the 32 mathematical tasks, Mike
implemented eight group discussion tasks, which were all car-
ried out through open dialogue, so that each group of students
could have an opportunity to fully understand the solution
process and results, as well as compare different solution me-
thods and representations. There were 17 tasks implemented
using teacher-student dialogue; however, 16 used closed dia-
logue as the interaction, and only one engaged in open interac-
tion when asking the students to observe the similarities and
differences between two similar images, at which time the stu-
dents stated their complete thoughts (M0410Ob). As for indi-
vidual practice tasks, other than two that used open dialogue in
interaction (M0417Ob, M0525Ob), four were simple computa-
tional calculations or word problems. After the students fin-
ished solving the problems, Mike asked one student to write his
solution on the board, and then closed dialogue was carried out
on the student’s solution. In addition, there was one practice
question that did not have any interaction because the time ran
out (M0313Ob). Of the 43 mathematical tasks, Yan imple-
mented three group discussion tasks, which were all carried out
through open dialogue, and another 32 were conducted using
teacher-student dialogue. Even though the two tasks on speed
provided students with the opportunity to fully explain the
problem (only one student answered, Y0415Ob), later, Yan
engaged in closed dialogue with the rest of the class based on
the student’s answers, so this was classified as closed dialogue.
As for the eight individual practice tasks, two tasks did not
have any interaction because the time ran out (Y0225Ob,
Y0408Ob), and five used more open interaction, including us-
ing operations to find the center of a circle (Y0318Ob) or ob-
serving the elements that make up cubes (Y0408Ob). Only one
was carried out using closed dialogue.
Characterizin g Tea chers’ Mathematics Instruction
Using Types of and Implementation of Mathematical
Through the analysis the types of mathematical tasks imple-
mented by the three teachers (high and low cognitive demand)
and their presentation methods (open and closed dialogue), it
was clear that the three teachers ustilized different types of
mathematics instruction, as shown in Table 4.
Table 4.
Mathematics instruction of the three case teachers.
Implementation Open dialogue Closed dialogue
Task type LianMike Yan Lian MikeYan
High cognitive
demand 25 10 9 0 10 9
Low cogniti ve demand 3 1 1 4 10 22
Note: Discarding tasks without teacher-student interaction.
For Lian, of the 32 tasks that had teacher-student interaction,
25 were high cognitive demand mathematical tasks with open
dialogue (including 24 group discussion tasks and one teacher-
student dialogue task, for 78% of the total). For three tasks,
although they were implemented with group discussion, they
were low cognitive demand tasks. For Mike and Yan, although
20 and 18 of the tasks were high cognitive demand tasks with
teacher-student interaction, respectively, the students only
learned 10 and nine tasks using high cognitive demand methods
(Mike had seven group discussion tasks, one with teacher-stu-
dent dialogue and two individual practice tasks, for 32% of the
total; Yan had three group discussion tasks, two teacher-student
dialogue tasks, and four individual practice tasks, for 22% of
the total). As for the rest, since the two teachers used closed
dialogue implementation methods, many open observation and
discussion or two-step high cognitive demand tasks had low-
ered difficulty due to closed dialogue and structural guidance,
and the students used low cognitive demand methods to learn
high cognitive demand mathematical tasks. On the whole, about
70% of these two teachers’ mathematics instruction allowed the
students to learn using low cognitive demand mathematical
tasks or to use low cognitive demand methods to carry out the
learning of high cognitive demand mathematical tasks.
Conversely, Lian did not use closed dialogue methods to im-
plement high cognitive demand mathematical tasks, and used
closed dialogue only for low cognitive demand tasks. For the
high cognitive demand tasks, only one was implemented thr-
ough teacher-student dialogues, in which the students were
asked to write word problems that corresponded to a × b + c × b
or (a + c) × b, and then carry out open dialogue to verify the
meaning of the equations (L0521Ob); the others were imple-
mented as group discussions.
In summary, based on the types of mathematical tasks im-
plemented, Lian and Mike used many high cognitive demand
tasks and Yan primarily used low cognitive demand tasks com-
bined with other implementation methods. It was found that
more than 70% of Lian’s instruction allowed the students to
learn high cognitive demand tasks with high cognitive demand
methods, while 70% of Mike and Yan’s instruction allowed the
students to learn low cognitive demand tasks or use low cogni-
tive demand methods to learn high cognitive demand tasks.
Lian’s tasks were primarily implemented as group discussions.
Mike and Yan used teacher-student dialogue as the primary
method of instruction in their implementation.
Conclusion and Discussion
The three case teachers presented mathematics instruction
that did not fully conform to the trends and policy assertions of
what mathematics instruction should look like in classrooms.
Lian was the most inclined toward conforming to mathematics
instruction policy recommendations, including adopting math-
Copyright © 2013 SciRe s .
ematical tasks with high cognitive demands, using open ques-
tions to interact with the students, and giving the students op-
portunities to think and express their understanding. Of the
mathematical tasks implemented by Mike and Yan, their in-
struction generally caused the students to learn mathematical
tasks with low cognitive demand methods implemented by
close dialogue. The different instructional methods of the three
teachers were similar to the research findings of Artzt and Ar-
mour-Thomas (2002) in an instructional analysis of 14 middle
school mathematics teachers in the United States. Three differ-
ent instruction types were found in that study. One was stu-
dent-centered, hoping that the students could achieve under-
standing in learning by providing mathematical tasks with suit-
able challenges. Another was teacher-centered, with an empha-
sis on procedural objectives and a focus on teaching the whole
curriculum, without providing enough time for the students to
express their own opinions or understanding. The final group
was mixed, which sometimes is oriented toward students, and
at other times is oriented toward the teachers. The trends and
recommendations for mathematics instruction have created a
beautiful picture for how the learning of mathematics can and
should take place, but in Taiwan, how many teachers are
teaching according to these reform recommendations? In the
future, it will be necessary to further identify and understand
the mathematics instruction of the majority of teachers in Tai-
wan, in order to serve as a basis for further exploration of how
curricular or instructional reforms influence student mathe-
matical learning.
Mathematical tasks are the core of student mathematical
learning, and they affect student understanding of the nature of
mathematics (NCTM, 1991). Thus, the past reform of mathe-
matics curricula has focused on the presentation of mathemati-
cal tasks (Stein, et al., 2007). However, this study found that
other than the types of mathematical tasks, a more crucial factor
could be how teachers implement mathematical tasks. This is
because even high cognitive demand tasks may be examined
and solved by students using low cognitive demand methods
because of the way teachers implement them in the classroom.
Although the results of this study are similar to the research
findings by Henningsen and Stein (1997), this study found that
the main reasons for students interacting with the mathematical
task types in low cognitive demand ways were because the
teachers frequently reduced the complex high cognitive demand
tasks into detailed closed questions with fixed answers, which
replaced the possibility of student analysis of the tasks and
decreased the opportunities for the students to explore and think
about the whole task. Is this a common method employed by
many teachers? This indicated that even though the type of
mathematical task is important, the method of implementation
is crucial. This study found that using group discussion to ad-
minister mathematical tasks generally required high cognitive
processing on the part of the students, while closed dialogue
tasks generally involved low cognitive processing. Does an inti-
mate relationship exist between the two instructional ap-
proaches? This should be the foundation for further exploration
of the relationship between the implementation of and the types
of mathematical tasks.
This study was funded by the National Science Council un-
der Grant No. NSC 97-2511-S-153-001. The authors would like
to thank the on-site teachers and assistants who participated in
this study. The viewpoints expressed in this article belong to
the author and do not necessarily represent those of the Na-
tional Science Council.
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