1
Supporting First Grader’s Ten-Structured Thinking by
addition problems
Karim Ezzatkhah
University of Illinois at Urbana Champaign
Literature Study
Introduction: Mathematics education, Developmental psychology, and Cognitive
psychology Researchers have done considerable studies in understanding of the variety of
ways children think about single-digit numbers addition and subtraction situations in the
last two decades of twentieth century. Researchers had considerable interaction in this
area in the United States (Romberg, Carpenter, and Moser 1982). Some other studies took
place during last fifteen years in different East Asian countries that; in their spoken
language are not any irregularities, as Japan, Korea, China, and Taiwan (Fuson 1992). In
this study I am going to review researchers findings in order to find out an appropriate
way to solve existing problem for English spoken language and other children who have
similar difficulties. For example in Iran (Persian speakers) children have the same
problem as English speakers children, instead in Turkey, Azerbaijan and Turkmenistan
children have not any problem by considering their speaking language.
Culture and Language: English words for two-digit numbers are irregular because
they are not named-value, for example 12 is not ten and two, or fifty seven is not five ten
and seven. In contrast to Chinese, Japanese, Korean, Burmese, Thai and Vietnamese,
words in which eleven is said “ten one” and forty five is said “four ten five” (Fuson
1990). Children in these countries seem to construct multiunit conceptual structures for
two digit numbers earlier than their age mates in English speaking countries. In the
United States, other study from Karen C. Fuson and her colleagues (1997) showed that
Latino low socioeconomic status children (Spanish Speaking) had better performance at
understanding two-digit numbers in the first grade than their English speaking children
with high socioeconomic status in the United States. Therefore it seems the high level
performance of East Asian countries first graders is not because of their language
advantages about number words. Spanish speaking children have similar problem in twodigit numbers word as U.S. first graders.
Karen C. Fuson and her colleagues (Mar 1992) have observed different
procedures in Korean children in solving word problems in the first semester of first
grade, before children have studied problems involving numbers large than ten in school.
1- Korean children had the rapid and accurate responses for word addition problems on
the sums to ten. 2-These children could solve single-digit addition problem with sums
between 10 and 18. 3-They could solve single-digit subtraction problem with minuends
between ten and 18. These findings imply to cultural differences in mathematics
education. Maybe Korean students have more support from their parents or Korean
teachers are well prepared. Many other social and educational aspects might have
influenced that researchers could not control. These children showed considerable
competency with all three kinds of problems (mentioned before), solving correctly 95%,
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85% and 75% of these problems respectively. Almost two third of the solution of the
problems above ten were addition and subtraction decomposition methods structured
around ten or known facts. Koran children did not have difficulty in two-digit numbers
between ten and nineteen as English speaker students. In Korean language eleven is ten
and one or twelve is ten and two.
Language structure: In most East Asian countries languages are structured around baseten system instead in English speaker countries, as U.S.A. is not. For example, in
Chinese, the number of 45 is four ten five and in English is called forty-five, that is not
clear as well as Chinese. Researcher found that, Asian language number words might
facilitate Asian children’s understanding of counting, number, and arithmetic (Miller &
Stigler, 1987- Miura 1987-Fuson 1992). I think the Asian children also have a little
difficulty, because when they call 36 as three ten six. It is possible to write 3106 instead
of 36. Over all language is likely 0ne of the several factors in the large differences in
mathematical achievement and development. Because Asian children outperform
American children in other field of mathematics that presumably would not be influenced
by the structure of number words. So, the structure of language is one of the many other
reasons to better performance of Asian children (Stevenson, Stigler, Lee, Chen, 1990).
Concrete modeling: Some researchers as Fuson (1992) argues that, Chinese, Japanese,
and Korean first graders given base-ten longs (tens) and units (ones) make multiunit
model, while American first graders model as units (unitary) not ten and ones (Miura,
1987). It is, because East Asian first graders work with tens and ones before attending
school, but American student’s work after going to school with tens and ones (Miura
1987). Therefore, It depends on children’s learning experience, and it is possible
overcome this problem in kindergarten for American children.
Fuson (1992) argues that, children in Asian countries are familiar with metric
system and abacus. These familiarities facilitate understanding number base-ten structure
and place-value; instead American children have not such experience. I think six years
old children in Asian countries do not have experience with these devices, unless, they
have been taught by parents or teachers in preschools and kindergartens. Thus, abacus
and metric system may not be as cultural advantages.
Teaching method: Teachers teach to East Asian first graders, when teaching single-digit
addition and subtraction by derived fact strategy. They also teach decomposing strategy,
when teaching single-digit additions. For example, in term of combining 5+6=, East
Asian first graders decompose 6 to 5 and 1; then, they say five plus five is ten and one
more or “ten one” (Fuson, Stigler, & Bartsch, 1988). American first graders have not
been taught in the same method. Therefore the differences perhaps are due to teaching
method, rather than other factors.
Children counted quantity ability in different cultures: Counting is the method used in all
cultures to differentiate and label quantities. So, children in each culture must learn some
strategies to count quantities (Fuson, 1992).
1. Children have to learn number sequence in their own culture.
2. Children have to learn how to point usually to each member of collection in
counting position in own culture.
3. Children have to learn indicating act to connect one number label to entire
collection.
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4. Children have to learn cardinal meaning in a collection (last label indicate the
total number of the collection).
5. Children have to learn the methods of remembering already counted collection.
The specific strategies used by children for solving addition and subtraction problems
depend on the culture’s counting system. Addition by young children in the United States
usually involves the counting of collections of objects (Steffe, Thompson & Richards,
1982). Researcher found that, Korean and Japanese children use the same strategy
(Ginsberg, 1989). Children in two different cultures use one specific counting strategy.
Children start object counting (cc) before coming to school, if objects are not available,
then fingers are often used as a substitute (Siegler & Sherager, 1984). Finger counting has
some limitations, for example when children want to solve the addition problem like:
8+9=? Using two addends with fingers in the same time is not possible. Therefore, using
objects would give more flexibility to model concretely two addends and count the
collections.
Children understanding of basic arithmetic: Jean Piaget (1965) after a lot of
experiments argued that children did not have a conceptual understanding of basic
arithmetic until the age 6 to 8years old. Piaget has divided children ‘s cognitive
development to four consecutive stages and he mentioned that children in third stage of
thinking development are able to understand basic arithmetic operations. However, more
recent studies, using different methods, showed that children have considerable
knowledge of arithmetic before attending school even from early childhood (Starkey
1992; Gelman & Starkey 1982). Research on human infants demonstrated their ability to
differentiate sets of one, two, and three objects from each other. This immediate
apprehension and labeling of small numbers has been termed subitizing. (Starkey, 1992 &
Wynn, 1992a). Children first cardinal meanings of number words are as labels for small
sets of perceived objects. These perceptual quantity meanings seem to be biologically
prepares (Fuson, 1992). Perceptual quantity provides an early basis to addition, ant it
continues to play a role in the more advanced conceptual level of addition and subtraction
(Jordan, Huttenlocher & Levine, 1992). Children by four and five years old rely on verbal
counting to solve simple arithmetic problems (Resnick, 1983). The use of counting to
solve arithmetic problems represents their informal knowledge, because children use
these skills without formal instruction. The most important finding across all of these
studies is that, children adapt their previous mathematical knowledge (counting skills) to
situation that, require addition or subtraction (Geary, 1996). Most kindergartners in the
U.S. have considerable experience by counting with a lot of variety of strategies (Gelman
& Gallistel 1978). Karen Fuson (1990) is called to this children’s counting ability,
counted quantity.
Addition and Subtraction Situations in the Real world: There are a lot of
studies about the kinds of word problems and whole number addition and subtraction
situations in the real world. Most of researchers have found-out recently that, there are
basically four addition and subtraction situations (Carpenter & Moser 1983, Fuson 1992).
When there are two quantities, children can combine them or compare them. Fuson
(1992) for combine and compare situations says Binary operations. When there is only
one quantity; children can add to that quantity or take from that quantity. These two cases
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have been called Unary operations. Distinguishing between two different operations that
the first is active and the other is static situation is one of the children’s problems in first
grades. Each addition and subtraction situation involves three quantities, any one of
which can be unknown. For example in 4+3=7 (equation) children have two addends and
they are obtaining the sum, in 6-2=4 (equation) children involve with first quantity and
compare with second quantity to get differences, or take away 2 from 6 (first quantity) to
find remainder. The symbols like +, - and = have different meaning in different situations
(Geary1996). For example, (-) has compare and take away meanings in two different
problem situation.
Children’s counting strategies: Children in the United States pass five different
levels of counting strategies during the development of counting strategy in early
childhood. The last level of counting is numerical counting that children involve with
abstract addition and subtraction (adding or subtracting symbols). In this level children
know each number as a combination of small numbers. For example 6 can be (1&5),
(2&4), (3&3), (4&2), and (5&1). They can also understand that, 7&4 can be 7&3&1, or
10&1. (Fuson, 1992). Researchers found that, children in the first grades have a lot of
addition, subtraction, and counting experiences from out of school. Teachers who are
ignoring these experiences, they create major problem to children who are eager to learn
mathematics. This ignorance of children previous knowledge would be problematic for
children in term of connecting their previous knowledge to new knowledge (school
knowledge) that, teachers are willing to teach. Researchers could found many different
counting strategies used by children when, they add collections. For example children for
solving simple addition problem like 2+3 use five general classes of strategies (Carpenter
and Moser, 1983).
 Using manipulative
 Finger counting
 Verbal counting (counting mentally)
 Derived facts counting (Derived fact strategy involves using memorized addition
facts)
 Fact retrieval counting (Children quickly produce the answer, without overt signs
of counting)
Most of children in the United States examine those strategies in out of school at in
their everyday life. Transition from one level to other is related to different factors, for
example, transition from finger to verbal counting is primarily dependent on the child’s
ability to mentally keep track of the numbers that have already be counted and those that
still need to be counted, and for most children shift from one level to other level is
gradual (Fuson, 1982).
Children’s Difficulties in identifying digits of Multi-digit Number:
Children in the United States receiving ordinary classroom instruction have considerable
difficulties constructing concepts of multi-digit numeration, addition, and subtraction.
The National Assessment of Educational Progress reported that, only 64% of third
graders could identify the digit in the tens place in a four-digit number, and less than half
identified the hundreds or thousands digit. This inability to understand or use multiunit
concepts affects multi-digit number algorithms (Brown et al., 1989).
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In a heterogeneous sample of fifth graders from the Chicago area, only 69%
correctly solved a three-digit subtraction requiring two trades (Stigler & Stevenson,
1990). Many errors that U.S. children make on numeration and on multi-digit subtraction
tasks indicate they interpret and treat multi-digit numbers, as single-digit numbers placed
adjacent to each other, rather than using multi-digit meanings for the digits in different
positions. Fuson (1990) has argued: “Thus, they seem to be using a concatenated singledigit conceptual structure for multi-digit numbers. The use of this concatenated singledigit meaning for multi-digit numbers may stem from classroom experiences that do not
sufficiently support children’s construction of multiunit meanings, do require children to
subtract multi-digit numbers in procedural rule-directed fashion, and do set expectations
that school mathematics activities do not require one to think or to access meanings.”
School textbooks present multi-digit numeration and subtraction in ways that interfere
with children’s ability to make generalizations (Fuson, 1988).
In the U.S.A. work on multi-digit addition and subtraction is distributed over four
or five years, while in Asian countries and Russia such work is completed by the 3rd
grade (Fuson 1990). Most previous work on children’s invented solutions has been
limited to two-digit numbers. Many of these procedures do not generalize well to large
numbers and/or they depend on special solutions for the decades. (Fuson, 1990)
Baroody (1990) has emphasized that, “many children in the United States have
difficulty learning place-value skills and concepts, and the teaching of these fundamental
competencies needs to improve.” He also argued; “there is no clear answer that how
much it takes time student understand multiunit concepts”. Currently research does not
answer what concrete or pictorial models need to be used to construct a secure and deep
multiunit concept (Resnick& Ford, 1981). Actually what we are going to teach in term of
multi-digit unit and what extend student need the concepts and procedures. Is not clear.
Instructional Materials: Educational researchers have shown a growing interest in
examining children learning of mathematics in situations that involve the manipulation of
physical objects and events (e.g.; Greeno1988). For example he has suggested that
manipulating the physical referents of symbols helps engage children in “meaningful
learning activities”. National council of Teachers of Mathematics (1998 Pp112) has
emphasized: “Technology can help children develop number sense through computer
manipulatives. For example students can break computer base-ten blocks into ones and
glue ones together to forms tens. The computer also links the blocks to symbols and so
forth”. Most researchers in the U.S.A use base-ten blocks as manipulatives. These
manipulatives are not appropriate for first graders, because need measurement skills. For
example first graders have to measure Long By Small Cubes and make sure that each
long is ten Small Cubes. Some researchers as Fuson and her colleagues (1997) have used
money as manipulatives to teach place-value concept. We should realize that money has
its own value in any places and its value is not based on the place, which we are putting
them. Therefore the best concrete objects are wooden or plastic sticks in ten centimeters
size, that children can count-up and make groups of ten. In computer screen tally marks
or countable objects are appropriate manipulatives to demonstrate two-digit numbers in
place-value system.
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Children Conceptual Structures for Two-Digit numbers: Developing
conceptual structure of two-digit numbers depends on some factors: 1- the language that
is used by the teacher and students and cultural issues. 2-The type of physical materials
that are used. 3- the problems that are to be used. 4-The structure of the class activities.
5- the student’s cognitive development. These components concert with one another to
support children construction of meanings at two-digit numbers. Through various
projects Karen C. Fuson and her colleagues (1997) identified five different conceptions
of two-digit numbers during their conceptual development of two-digit numbers “
unitary, decade, sequence, separate and integrated”. Children Initially have unitary
conception of any discrete quantity.
One…. Five
*****
(Quantity)
1-9
Five (Number word)
(number mark) 5
Figure 1-Unitary Single-Digit
Children in the first grade have the same conceptual structure for big quantities. For
example 15 is 1,2,3,……..14,15 units (Fuson1997).
One,….fifteen
*****
*****
*****
Fifteen
15
Figure 2- Unitary Multi-Digit
This unitary multi-digit conceptual structure has to transit to separate Tens and Ones,
then, sequence Tens and Ones and finally integrated separate-sequence Tens and Ones
(Fuson1997).
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One ten and five ones
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
One ten five ones
15
Or fifteen (the name 0f number)
Figure 3- Multi-unit concept of two-digit number
The main different between figure2 and figure3is that, children in the figure 3 have to
learn:
1- counting the quantity, grouping by ten and ordering tens and ones from left to right.
2- Learning relationship between ordered quantity and English number word.
3- comprehending relationship between ordered quantity and symbols, that is in the same
order.
4-Making relationship between English word and ordered symbols.
These two figures consist some similar activities that students have learned before.
Thus, this knowledge will help them to accommodate the new concept (multiunit
conceptual structures) based on their previous information.
Present Textbooks and Teaching/Learning Two-Digit Numbers:
In the United States, elementary schools children learn two-digit numbers from
second grade. Reading and writing two-digit numerals is initially based on multiunit
conceptual structure. Children learn addition/subtraction of the most difficult single-digit
sums between ten and eighteen (e.g.; 7+8=15 and 15-8=7) after learning two-digit
number without trading as (56-21) and (34+23). These teaching and learning processes
and a lot of difficulties that have been mentioned, has created existing problems in
teaching and learning place-value system and addition subtraction problems in the early
years of elementary schools.
What is the problem?
Many first graders in the U.S.A. do not understand the base-ten structure of multidigit number words (Fuson, 1990).
U.S. students have problem with grouping quantities and understanding placevalue system (Miura, 1987).
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There are irregularities in reading and writing two digit numbers. Thus, first
graders need time and effective instruction to learn those numbers.
Reading and writing multi-digit numbers is difficult for children because written
numbers do not correspond directly with spoken counting numbers, for example fifteen is
not one ten and five or thirty seven is not three tens and seven.
Children’s counting knowledge from out of school is not like counting procedure
that children learn at school. For example, children when count a collection, they get the
result as a combination of ones (unitary concept), in fact we are expecting the result in
base-ten system (multi-unit concept). ( Fuson, 1992).
Place- value and base-ten knowledge
Children initially do not realize that our number system is place-value system; the
position of a digit determines its value. For example in written numbers as 5432, 3542,
3452, 2345, digit 5 represent different values by virtue of its position.
Children come to school with a counting–based concept of numbers, and do not
know about grouped items. For instance they look at 14 as 14 units, not one ten and four
ones. So, children should discover, first the importance of grouping collections into larger
units or multi-units concept. Second, children have to understand and examine grouping
by ten and, ordering groups from right to left, and from smaller to larger groups. Third,
children must reinvent place-value rules in written numbers. Fourth, nominating digits
from left to right to read written numbers. Fifth, getting mastery in term of trading from
larger digits to smaller and vice versa, for example they have to discover that, two tens
and four can be represent as, one ten and fourteen ones, or twenty four ones. Therefore,
children understanding of place-value and base ten system is necessary for some reasons:
First, children understanding of the conceptual meaning of spoken and written multi-digit
numbers is dependent the knowledge of base-ten system. Second, the understanding that
multi-digit numbers represents groups of ones, tens and, ten times ten, and etc (Fuson,
1990).
Instructional Model of Two-Digit Numbers:
In Fuson’s model, children’s conceptual structure about single-digit numbers
initially is unitary. They have to develop this model gradually to multiunit conceptual
structure. She and her colleagues have suggested five different stages (unitary, decade,
sequence, separate and integrated) in development processes (Fuson et, al 1997). We
know children in the development processes have to accommodate the new knowledge
that is based on their previous knowledge Piaget (1965). Therefore children would extend
their Triad model Fuson (1997) to a different model as an instructional loop proposed in
this project (Figure4). This model has different stages.
The first stage contains 45 basic facts problems that, students will solve by their
own strategies through a mathematical game. Students follow this instructional model by
teacher’s guideline to transit from unitary conceptual level to multiunit conceptual level
of thinking structure.
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Single-digit addition, sums between ten and eighteen
As (9+9) by appropriate manipulitives
Conceptions of the quantity
******
******
******
English word (eighteen)
Symbols (18)
Unitary
conception
Counting on the quantity and grouping by ten
Ordering from left to right and from bigger to smaller
(Groups of tens and ones)
* *
* *
* *
* *
* *
* *
* *
* *
*
*
Tens/Ones
1
8
18
Multiunit Mu
conception
Figure-4. A developmental sequence of conceptual structures for two-digit numbers
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Purpose of this study
It proposed, that the students those learn two-digit numbers by addition of the
most difficult single-digit sums to 18, with unitary conceptual structures. And, they use
appropriate manipulatives or computer to model the learning situation. These students
will be able gradually acquire multiunit conceptual structures and understand two-digit
numbers sense. They also learn difficult single-digit addition sums to 18, with a deep
understanding. Therefore, their understanding of two-digit numbers and single-digit
addition sums to 18, would be better than students that are following textbooks based
instruction in this field.
Methodology
1- Participants and settings:
This project will take place in one of the Champaign-Urbana elementary public
schools. All intact class students will participate in the project. First I will study student’s
files to obtain certain information as: age, gender, socioeconomic status, cultural
background and mathematics achievement tests (teacher made). If they had IQ scores or
some similar information; It will be considered as one of the selection criterion. I will
choose three identical classes, two classes will be experimental and a control groups.
2- Teachers training
Three first grade teachers will participate in this study. The teachers who are
teaching in experimental classes would have short term training to master the
investigative approach and instructional model of this study (maximum 8 hours).
Teachers would have a short training time about running on the preliminary test, pretest,
posttest and also interviews (2 hours). Teachers training will be in four phases:
Phase One: In this part teachers will be familiar with the preliminary test and testing
strategies. I will emphasize to them to consider this knowledge as baseline, so they will
be recommended to teach and complete student’s knowledge in these terms with their
own strategies. So it will be possible to have remedial instruction after getting test results.
In this phase teachers will be aware that their students in all three classes would have
equal knowledge in understanding at number sense based on preliminary test content.
Phase two: Before intervention all students will pass pretest. So, teachers would
have a short period of time to be familiar with test and its administration.
Phase Three :In this part, the experimental teachers who had training would learn
about the intervention phase of the project, manipulatives and applying investigative
approach in classroom activities practically. Only one experimental teacher will be
trained to use Applet in her class. The other experimental teacher will be trained to teach
with manipulatives.
Phase four: In this phase of the training teachers will participate to be familiar with
posttests and testing strategies. This orientation takes place only after intervention.
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3-Design
This study has been designed as three nonequivalent control group design with
pretest, posttest and supplemented by interview. The treatment in this study is: (teaching
via Instructional model of two digit numbers that, has been scheduled for this study). The
experimental groups have treatment beside its traditional instruction; and they learn by
two different concrete modeling (one class by computer assistant and other class by
discrete materials). The control group has only traditional instruction.
Length of one session mathematics class in first grade (one hour) in a day:
1- Traditional curriculum for control group
60 minutes
2-Traditional curriculum and treatment for experimental groups
40 minutes
20 minutes
Table1shows the operational timeline of the research project at 21 school days.
Diagram of nonequivalent control group design:
Dependent variable
O
Students performance
In pretest
Treatment
X
Twenty
teaching
Sessions
By computer
Class B:
Student performance
In pretest
Twenty
Student performance
teaching
in posttest
Sessions
By
Manipulative.
Class C:
Student performance
In pretest
Non
Class A:
Dependent Variable
O
Student performance
in posttest
Student performance
in posttest
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4-Experimental Procedure
Preliminary test: preliminary test includes fifteen questions needs 30 minutes. The
preliminary test will be administered in one session by teachers and will be scored by
researcher. Each question has ten points.
Preliminary test includes:
1-Counting skills:
1-1-Oral counting
1-2-Skip counting
1-3 Comparing quantities
2- Understanding the meaning of number:
2-1 Cardinal meaning
2-2 Ordinal meaning
2-3 Measurement meaning
2-4 Nominal meaning
3- Logical thinking Abilities:
3-1- Equal partitioning
3-2-Grouping
3-3-Ordering
3-4-Number Relationship
4- Number of 36 Single Digit Additions and 36 Subtractions Sum to Ten.
5- Number Literacy:
5-1-Numeral Recognition
5-2-Numeral Reading and writing
Preliminary test for experimental and control groups include tests from all five
categories (above mentioned). Students who can answer at least eighty percent of the
tasks correctly, will be allow to participate in both experimental and control groups. Preexperimental instructional phase will conduct to make group’s prerequisite mathematics
knowledge identical. Thus, Teachers may conduct or provide remedial instructions to
meet the 80% criterion before experiment.
Pretest &Posttest: Pretest includes ten questions and takes time at least twenty
minutes and will administrate by teachers. Posttest has twenty questions (ten questions
the same as pretest and ten more) and will administrate by teachers. Therefore pretest has
fewer questions than posttest, but pretest’s questions are key questions about in all four
domains as below. Each question in pretest has ten points and in posttest five points.
1- Evaluating student’s knowledge in solving Single-digit addition problems that
sum-up to eighteen
2- Composing and decomposing two-digit number in first decade of number
sequence and distinguishing between values of the digits and their order
3- Students understanding of the place-value system in writing two-digit numbers
at the number sequence (prediction from tasks)
4- Producing other two-digit numbers more than 18, like19, 20,21…. etc.
understanding that some numbers has different name (for example two tens is
nominated twenty)
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Interview: At least five students from each group will be interviewed by the teacher to
figure out students’ strategies in solving single-digit addition problems and value of digits
in two-digit number. The interview will take place in the classroom in an appropriate
time. The interview consists at fixed closed ended questions for all participants from both
experimental and control groups. The teachers may write down students’ answers or
audiotape (in case of having parent’s permission). The researcher will do coding,
quantifying and interpreting students’ answers. The main question of the interview would
be:(How do you come-up to this answer). The teacher will ask the question by showing
student’s worksheet at pretest and posttest.
Manipulatives: This project is designed to work with two different manipulatives.
First will be concrete objects as:
1- 100 wooden or plastic sticks.
Tens Ones
2-Rubber bands.
3- a piece of paper illustrated the place-value mat.
Second by computer assisted program: URL to computer program is:
(http://www.mste.uiuc.edu/users/mmckelve/applets/Counting/done/counting.html).
Treatment: The experimental and control classes will continue their normal
instructions. The experimental groups have new instruction procedure beyond their
everyday mathematics courses. This instruction includes learning 45 single-digit addition
problems (Figure 6) and regrouping the results as ten’s and ones (Figure 7) by concrete
modeling (discrete objects or computer assisted).
Process of the treatment: The instruction will take place during sixteen short
sessions and each session will be only twenty minutes (Table 1). The teaching day and
time will depend on the teacher’s agreement. In the teaching time the researcher may
attend the sessions upon the teacher’s agreement or the teacher make report the class
activities for the researcher. The researcher would give instructions for everyday teaching
procedure in a short meeting with the teacher or by E-mail memo to coordinate the
teaching processes. The processes by computer and concrete manupulatives are
illustrated in the next page (Figure 7).
+ 1
Figure6- Addition Chart
1
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
3
4
5
6
7
8
9
4
5
6
7
8
9
5
6
7
8
9
6
7
8
9
7
8
9
?
8
9
9
14
Pick up tallies and add:
And
Is
Tallies and
7
+
tallies are
5
=
12
tallies
12
12 is:
And
Ten
12 are
two
ten and
ones.
Tens Ones
1
2
12
Twelve
Figure 7-The processes of instruction by computer assistant and concrete
manupulatives
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Instructional Games: The game has two parts based on processes that, has
illustrated in (Figure4). The students can play individually or in small group by teacher
guidance.
First part: In this part of the game students explore 45 additions as illustrated in
Figure5. Each student has addition card and other manipulitives as indicated. For
example, teacher asks students to look at the addition card and find-out the sum of
7+5.The students by their own strategies will try to figure-out the sum of 7+5 and write
that, in the right place on the card. In this particular example the sum is 12. Supposing all
students can indicate English word (twelve) but it is possible some of students got trouble
to writ 12. The teacher without any explanation should help them to writ 12 as a single
digit number with two adjacent symbols. In the first grade the students have similar
experience in learning English words, for instance they know that (S) and (H) have
different sounds, but when they come together as (SH), It has different sound.
+
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
10
2
3
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Figure5-45 additions sums ten to eighteen
Second part: In the first part of the game students learn all difficult single-digit
additions by unitary conceptual structure. This knowledge helps them to examine
multiunit conceptual learning gradually, and to understand digit’s value in two digit
numbers. In this part of the game students use their findings (sums) to perceive the value
of each digit in two digit numbers. The game starts with one addition problem like
7+5=12, then teacher lead students to make collection of ten and pack it by rubber band.
In this example students have one set of ten and two more sticks. The teacher lead
students to arrange findings from left to right and from larger to smaller. After
arrangement, the students have to writ number of tens and ones in place-value mat. They
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read one ten and two. The students make connection between previous knowledge (12)
and new knowledge (0ne ten and two ones). This process continue until, to figure-out
relationships between two digit numbers in the first decade of the number sequence.
Tens ones
1
2
12
Twelve
Understanding place-value system needs multiunit conceptual structures.
Therefore children initially start by unitary conceptual structures and gradually would
learn digit’s value in the decades and the numbers between decades until one hundred at
the first grade.
It is possible to develop this game on computer screen by individual student. If students
play this game by computer assistant, they should share their findings with peer students
in small group.
Three main aims would be considered in this process. First give opportunity to
children to produce mathematical concepts by own strategies. Second perceive their
products as well. Third, predict the use of mathematical concepts and procedures, which
have been learned in the appropriate problems. Therefore three crucial goals in any
mathematical course would be: producing, perceiving and predicting.
Supplementary game: In order to learn the value of digits in two digit numbers,
children play another game. Student’s individual using computer can play this game, or
they can play in a group of two students using the manipulatives. Manipulatives are the
same as mentioned before. The main aims of the game are producing two digit numbers,
predict their production’s name, and perceive the value of each digit. The rules of the
game are: 1- counting-up to ten. 2- grouping by ten. 3- ordering groups (tens and ones).
4- writing the number of tens and ones in the place-value mat. 5- Nominate the numerals
that, is the winner. 6- one’s winner is that can produce and predict a higher number of the
writing and reading skills of his/her products, and demonstrate deeply understanding by
composing and decomposing the numeral digits in limited time.
Children who can clearly explain the value of each digit at two digit numbers,
means that they are thinking in the multiunit conceptual level. These students could
transit from unitary level to multiunit level of thinking by using instructional model of
this project.
5-Measure of Skills and Understandings
All students will have preliminary test, pretest, and posttest, some will have
interview to figure out their understanding of addition problems and transition from
unitary conceptual level to multiunit conceptual structures.
Children’s single digit addition and subtraction tasks : All participants must
know 36 single-digit addition and 36 single-digit subtraction problems that sum up to ten
(e.g. 3+5=8 & 8-3=5) before treatment. Therefore one part of the task should cover these
problems. The next part of the task consists at single-digit addition problems sum up
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from ten to eighteen (e.g. 7+8=15). Children would have opportunities to choose
strategies that they prefer and know such as mental calculation or using fingers and other
manuplatives.
Children’s understanding of two-digit number: Students have to be able read and
write two-digit numbers particularly in the first decade of number sequence. The task
should measure the students ability distinguish the value of digits in two-digit numbers.
These tasks include composing and decomposing each two-digit number to tens and ones
(e.g.25 is two tens and 5 ones or one ten and fifteen ones). Two word problems as a
sample of word problems in this study are presented as below:
Problem1: I was looking outside and for each car that passed on the street, I got a stick,
and these are the sticks that I collected in a small amount of time. Can you tell me, how
many cars passed on the street?
Problem2: I counted the number of first graders, as they entered to the class. There were
sixteen students. If I grouped the students by ten, how many ten and how many left over
will I have?
The first problem measures the student’s thinking of the unitary conceptual
structure. The second problem would measure their understanding of the multiunit
conceptual structure. The students may apply different strategies to solve these problems.
Measuring the process of thinking: An individual interview will carry out to
assess a child’s understanding of addition and place-value concepts. In the interview,
children will be asked to explain their solution strategies and their base of thinking about
numbers sense.
Scoring policy and validation: To acquire credit, the child has to indicate the number
of tens in written numeral (two-digit number) correctly. He/she also has to indicate the
number of ones (left over) in written numerals correctly, and mention the English word as
the name of the numeral. The child has to analyze two digits in two-digit number and
rename them as tens and ones (e.g. 25 is two tens and fife ones or one ten and 15 ones or
25 ones). In the addition problem two important points are: (a) correct calculation of the
sum by any individual valid strategy by verbal explanation. (b) To write the numeral and
mention the English word.
Test-Retest Reliability
To make sure that pretest and posttest measure as well as possible, I have chosen
all testes from ERIC database from the same domain that the instructional content of this
study follow-up. Tests have been modified to best fit to the content of the course.
Validity
Pretest and posttest have been designed to measure exactly children understanding
of place-value system and single-digit addition problems. Validity is my primary concern
in term of gathering data.
Remedial Instruction: After collecting data, I will support control group’s teacher to
teach to the control group the same as experimental groups. So all three classes will get
benefit from new strategy of teaching two-digit number and single-digit additions.
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Results
Data will be gathered after preliminary, pretest, and posttest. There are several
acceptable ways in which the data associated with a multi-group pretest; posttest design
can be analyzed statistically. First, I use the analysis of covariance to compare the various
groups in terms of posttest means, after these means have been adjusted to account for
any differences that exist among the groups on the pretest. Second, I compute a
difference (i.e., gain) score for each subject by subtracting his pretest score from his
posttest score, with one-way ANOVA applied to the gain scores to see whether the
groups differ in terms of the mean gain from pre to post. Third, to find out main effect of
groups, main effect of trials, and groups-by-trials interaction, I may use Lindquist Type 1
repeated measures analysis of variance. Fourth, I will apply a Kruskal one-way analysis
of variance of ranks to the difference scores. Of course, an overall significant finding
from any of these statistical approaches would require a follow-up by an appropriate
multiple comparison investigation to find out exactly which groups are significantly
different from one another.
The preliminary testing will identify the first graders who can count-on orally at
least up to twenty, and object counting up to ten correctly. This testing also will indicate
their mastery to the 36 single-digit additions and subtractions sums to nine. This
Knowledge would be the base for treatment of single-digit addition sums to 18 and
understanding place-value concept and procedure of writing and reading two-digit
numbers.
Children’s single-digit addition sums to 18: Children’s single-digit
addition and subtraction methods become more complex, abstract, internalized,
embedded and abbreviated. Several development levels in these methods have been
identified, and this progression of methods can be described at variety levels of
specificity (Fuson1997). In this project, the most important point is children’s conceptual
structure of two-digit number in the first decade of the number sequence. Children’s first
level of thinking about sum of additions; is relevant to unitary conceptual structure from
two-digit numbers. They understand each numeral as single-digit number. Children also
nominate them and write by two adjacent symbols. One part of the posttest will indicate
their correct learning of addition problem.
Children’s conceptual structure of two-digit number: In the second level,
children’s unitary conceptual structure of two digit numbers will change smoothly to
multiunit conceptual structure of two digit number. For example; after calculating each
sum (7+8=15), they decompose 15 to ten and five. This new relationship between whole
(15) and parts (10and 5) is in one hand another addition (10+5=15), and the other hand
includes a new arrangement of two parts (ten and Five). Children actually examine
grouping by ten, and ordering groups of tens and left over as groups of ones in cognitive
domain. The second part of posttest will indicate this processes of children’s thinking
evolution of perceiving numerals.
Children’s invented procedures: There is no doubt that children employ a lot
of methods and strategies in the process of transit from unitary to multiunit conceptual
structure. The interview with students will indicate the different strategies. I suppose,
19
they will use a lot of sophisticated methods to model addition problems and analyze sums
to tens and ones. I will classify the methods children use in learning the processes:
methods to add two addends in single-digit addition problems, decompose tens and ones
method, methods of demonstrating tens and ones and possibly other methods, which they
use in the processes. I expect children’s previous knowledge help to promote the
classroom activities, and give opportunity to teacher to lead them through the
instructional model used in this project.
Discussion
This study is going to address the documented need for the development of
models of instruction to foster and assess children’s growth in multi-digit number sense
during the first grade. Young children invent increasingly sophisticated and more
efficient strategies in terms of solving their mathematical problem (Baroody 1999). This
study would create opportunities for children to think and discover strategies by concrete
modeling to learn two-digit numbers.
On all tests and interviews measures, the performances of first graders who learn
the instructional model of this project have to exceed those learned by traditional
approach based on textbook. The successful instructional model will indicate that
appropriate manipulatives (different from ten blocks) facilitate understanding of placevalue and two-digit numbers. Therefore, the effects of manipulitives are considerable
particularly in transition from unitary conceptual structure to multiunit conceptual
structure. The results of study will indicate that, teachers should consider children’s
previous knowledge in meaningful learning of mathematical concepts and their interest to
study mathematics. The instructional model of this project will be generalized to similar
situations. This study will indicate that, children invent a lot of strategies to solve
mathematical problems and they conceptualize place-value system by their preferred
way. Thus, teachers can squeeze the mathematics content in between the students
Knowledge at cognitive domain.
Analysis
Some assumptions for this study are proposed: 1- this study can help children to
learn place-value system in first grade, because of their transition from unitary to
multiunit conceptual structure levels and also, children learn part-whole relationships
between digits, in two-digit numbers. It seems children previous knowledge about
counting-on by base-ten system facilitates students understanding of place-value concepts
and procedures. 2- this study can help children to reduce systematic counting errors
(Fuson 1999). 3- this study can help children to reduce trading and renaming errors that
students in elementary schools have in multi-digit addition and subtraction. 4- children
have opportunity to discover counting rules such as cardinality, ordinality and numberafter rules. 5-Children have chance to compare numbers and learn number size (Baroody
1999). 6- appropriate manipulatives have basic rule in teaching and learning processes.
According to Jerome Bruner (1964) transition from inactive phase to pictorial, need
precise instruction. This study is not going to evaluate the transition of learning to a new
situation. This is limitation of study and further research may clarify this issue.
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The major limitation of study is that their designs do not permit an evaluation of any of
specific features of the instructional model. These features stem from the need to provide
children an opportunity to construct multiunit conceptual structures of mathematically
different English named-value system of number words and the positional base-ten
system of written marks and to think about how these systems work in two-digit numbers
reading and writing. It means evaluation of student’s path from unitary to multiunit
conceptual structure of two-digit numbers construction; will not be possible. The next
limitation of the study is my assumption of student’s readiness of logical knowledge and
ability to classifying and ordering activities in cognitive domain. Thus, it is possible some
of student lack of these experiences. Selection is a threat in any Nonequivalent control
group design, because there is no random assignment. Because of selection problem,
maturation and history naturally will be problem. Mortality, and regression also can be
problem in this study.
The results of this study will be recommended to National Council of Teachers of
Mathematics to consider in its annual session in order to make change in the instructional
program.
21
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