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Copyright ? 2006-2013 Scientific Research Publishing Inc. All rights reserved.
2013. Vol.4, No.7A1, 23-32
Published Online July 2013 in SciRes (http://www.scirp.org/journal/ce) http://dx.doi.org/10.4236/ce.2013.47A1004
Copyright © 2013 SciRes. 23
A Mathematically Creative Four-Year-Old—What Do We Learn
Mathematics Education Department, Kibbutzim College of Education, Tel Aviv, Israel
Received May 21st, 2013; revised June 21st, 2013; accepted June 28th, 2013
Copyright © 2013 Ruti Steinberg. 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.
A 4-year-old child, who is very interested and precocious in mathematics, was interviewed doing mathe-
matical tasks in order to find out how advanced can a 4-year-old child be? His mathematical knowledge
and ability are very high. Danny was able to count objects and add them, memorizing many of the addi-
tion facts. He was able to check if numbers are even or odd. He solved a variety of challenging addition,
subtraction and multiplication problems. He could read and write large numbers in hundreds and thou-
sands and could add large numbers. Analysis was done on the kind of problems the child was able to
solve, their level of difficulty and the solution strategies the child used in light of what children usually do
to solve these problems in ages 5 - 8 (Carpenter et al., 1999). Danny also showed creativity, including in-
venting problems for himself to solve and finding mathematical situations in his environment to attend to.
An analysis was done on the creative components of his solutions and problem posing using the literature
on creativity and creativity in mathematics learning (Leikin & Pitta-Pantazi, 2013). Interview with his
mother showed that she supports Danny’s mathematical development by being responsive to his initia-
tions and allowing him to explore his ideas autonomously. Her support was discussed in light of different
support patterns of parents and what kind of support is especially beneficial to the child and encourages
creativity (Leder, 1992). I discuss implications for education with an emphasis on what kindergarten and
school can do to promote problem solving and creativity in mathematics.
Keywords: Creativity in Mathematics; Mathematics Education; Early Childhood; Precocious Child in
This paper will explore the mathematical world of an unusu-
ally talented 4-year-old child (hereafter Danny, not his real
name). There is little documentation of mathematically talented
children at such a young age so it is valuable to see what we
can learn about how this child does mathematics. Danny is a
creative and inventive problem solver. I will use his rich mathe-
matical world as a take-off point to discuss the importance of
creativity in learning and teaching mathematics. I will also re-
view problem solving in mathematics and its implications for
instruction and for enhancing creativity. I will review studies
that show that even very young kindergarten children invent
creative solution strategies.
The demands of life in the 21st century will require an ability
to apply mathematical thinking in new and changing environ-
ments. Children will need to use mathematics flexibly and crea-
In the interview with Danny I asked him to solve a variety of
challenging word problems. I will review and analyze the
structure of the problems, their difficulty level and the kind of
solution strategies young children typically use to solve them.
Some of the review will be given in the section on theoretical
background. Some analysis will be in the results section, where
each problem that I asked Danny to solve is presented with an
explanation of its unique features that will facilitate comparison
of Danny’s solution methods to what young children usually do.
Since Danny solves problems that usually only older children
solve, I will compare his solutions to the known strategies of 5 -
When confronted with an unusual child like Danny, it is in-
evitable that we ask questions about how he reached such an
advanced mathematical level. What interactions between Danny
and his family have helped him develop his mathematical
thinking? To provide necessary background for this topic, I will
review studies about styles of parent-child interactions at home
that might enhance or hinder the child’s creativity and interest
of exploring mathematics (in the discussion section).
The goals of the study are:
1) Finding out how advanced can a 4-year-old child be in
solving mathematical tasks and what kind of knowledge and
strategies the child can develop at this early age;
2) To see if a 4-year-old child can express creative behavior
in solving mathematics tasks;
3) To determine what personal traits of the child and what
kind of supporting environment allow the child to become so
advanced in learning mathematics at such an early age and to
show mathematics creativity. To draw conclusions about creat-
ing supporting environments for children so they can learn
mathematics creatively and meaningfully in kindergartens and
Creativity in Mathematics Learning
Only in recent years there is an emerging research on crea-
tivity in mathematics and not just on creativity as a general trait
(Leikin, 2009a). Summaries of this research can be seen in a
book (Leikin, Berman, & Koichu, 2009) and in a special issue
of a journal (Leikin, Pitta-Pantazi, 2013). There are many defi-
nitions of creativity. I will use a definition of creativity by Tor-
rance (1974) that has been used to build tools to identify crea-
tivity and is common in the recent studies on creativity in
mathematics (Leikin, 2009a). Torrance suggested four compo-
nents of creativity: fluency, flexibility, originality and elabora-
tion. Fluency refers to a flow of ideas, “not getting stuck”.
Flexibility is related to coming up with different ideas, finding
more than one solution or solution strategy. Originality means
having a unique idea or a solution that is rare or that others
haven’t thought about. Elaboration means taking it further—
framing a more general or abstract idea or integrating ideas and
taking them to the next level. When people commonly think of
creativity they think of only the component of originality—
producing different product or process than usual. We see in
Torrance’s components that there is more to creativity than ori-
ginal ideas. This is especially useful in education and can give
tools to advance children towards becoming more creative. In
order to be original you also need flexibility and fluency (Lei-
kin, 2009b). In order to produce an idea that is different than
others, it is also needed to be different from what you have
learned or from your previous ideas.
Milgram and Hong (2009) criticize the common ways to
identify gifted children, in many countries, for a special support.
These tests are based, many times, only on logical thinking and
IQ. Thus, they miss creative children and do not give them the
chance to develop their potential fully. Milgram and Hong dis-
tinguish between “expert talent” and “creative talent”. Expert
talent is based on knowledge in the specific field that was ac-
quired by years of studying and working in the area. Creative
talent reflects the ability “to produce ideas that are imaginative,
clever, elegant, or surprising, beyond analytical thinking” (p.
Leikin (2009b) and Leikin and Lev (2013) consider creativity
in school students as “relative creativity” as it is usually related
to a new solution for the student for a problem he or she hadn’t
seen before or to produce original solutions to previously
learned problems in the context of the local learning mathemat-
ics in school. This is as opposed to inventing new ideas that no
one in the world had thought about as professional mathemati-
cians do. Professional mathematicians need to be expert in the
previous knowledge, procedures and techniques in mathematics,
but they also need to be creative and to connect and integrate
ideas, to ask new questions, to use intuition, imagination, and
inspiration so they can come up with new knowledge (Ervynck,
Tabach & Friedlander (2013) studied creativity of groups of
strong students in mathematics from grade 4 to 9 in the same
school. The children studied part of their school lessons in
separate small groups. They found that, in general, for all three
components of creativity, fluency, flexibility and originality,
the scores increased with grade level. There was a decrease of
creativity in the eighth grade when the students mainly used
algebraic equations to solve the problems, but in the ninth grade
the students used a more balanced mix of algebra and other
solutions. These findings led the researchers to assume “that an
increase in mathematical knowledge (i.e. grade level) has the
potential to raise the level of creativity as well—with possible
exceptions because of the temporary influence of learning a
new domain (in our case, algebra). Thus, the observed increase
in creativity scores throughout the upper elementary school
(Grades 4-6) can be attributed to students’ increasing familiar-
ity with the arithmetical domain” (p. 238).
A few studies found correlation of achievement and creativi-
ity among first to fourth graders (Bahar & Maker, 2011) and
first to fifth graders (Sak & Maker, 2006).
Can Creativity in Mathematics Be Developed?
There is evidence that it is possible to develop creativity in
all students (Sheffield, 2009; Hershkovitz, Peled, & Littler,
2009). Silver (1997) considered problem solving and problem
posing as main tools for the development of the components of
mathematical creativity in all students. Fluency can be devel-
oped by generating multiple mathematical ideas, multiple an-
swers to a mathematical problem (when such exist), and ex-
ploring mathematical situations. Flexibility can be enhanced by
generating a few mathematical solutions. Originality can be
helped by having children look at many solutions to a mathe-
matical problem and encouraging them to come up with differ-
ent solutions. Many educators and researchers suggest that crea-
tivity is a skill that can be developed. It is important to help all
students develop their potential by encouraging creativity from
an early age. If we develop creativity even for some of the chil-
dren we can increase the potential for them to develop in the
future new ideas and technologies in science and society.
Efforts should be made in schools to create conditions that
will promote creativity for all children. Special attention should
be paid to the kind of tasks that are chosen and to educate
teachers on ways to promote creativity. Hershkovitz et al. (2009)
suggested criteria for good tasks that can encourage school
children’s creativity. They emphasized the importance of work-
ing with children who are at different levels so a good task
should allow children to operate at different levels and sophis-
tication. Other criteria they suggest for such tasks are: “Enables
multiple solutions; Has different answers or different solution
methods; Is challenging even if it can be solved in simple ways;
Can be extended by further questions, ··· Enables generaliza-
tion and abstraction; Encourages investigation of different cases;
Encourages discussion and argumentation; Encourages the use
of deep mathematical principles” (p. 259).
Hirsh (2010) suggested to incorporating art activities in ma-
thematics and to explore mathematical, especially geometrical
aspects of art in the regular mathematical classes in school to
increase creativity among students.
Levav-Waynberg & Leikin (2012) conducted a teaching ex-
periment with high school students to find out if it is possible to
develop creativity in geometry. They worked with regular-high
and top-high level classes. They checked the students’ creativi-
ity before the experiment on 3 components: flexibility, fluency
and originality. An original solution was determined according
to its rarity in relation to all the solutions the students gave.
Copyright © 2013 SciRes.
Before the experiment, many of the students were not able to
solve the problems and if they solved them, they usually found
one solution. After the teaching experiment all students’ flexi-
bility and fluency increased. It was measured by the ability to
solve with different strategies and to give different solutions.
The top-high students’ measures increased more than the regu-
lar-high students. At the end students were able to solve more
problems correctly and many of them gave 3 solutions or solu-
tion strategies to the problems. The originality measure actually
decreased at the end. This surprising result is explained by the
researchers by the fact that many students now solved in a few
ways so it was difficult to come up with a unique solution. The
few students whose originality increased were gifted children.
Creativity in Kindergarten Children
It is reasonable to assume that some components of creativity
are not developing easily among young children. This is espe-
cially so with the flexibility trait. Young children are known to
need to solve problems exactly according to the structure of the
problem in “direct modeling” strategies using concrete objects.
Their thinking is usually not flexible. For example, they do not
see the relationship between addition and subtraction and do not
use such a relationship in solving problems (Carpenter & Moser,
1984; Carpenter, Fennema, Franke, Levi, & Empson, 1999).
Even so, there is evidence that even kindergarten children
can develop components of creativity. Different from the older
children, it seems that many young children are strong in the
originality trait and it is not so rare. I am referring to originality
here in the sense of being able to invent new solution strategies
and new ways to solve problems. These ways have not been
taught to the child. The criterion I am suggesting here is the
child’s ability to invent a new solution strategy or procedure,
not the criterion that checks if the strategy is different than
other children’s strategies. We know that certain kind of strate-
gies have been invented by many children. But each child was
able to invent himself. It is important to build on this creative
thinking of young children and not to direct them to believe
later in school that there is only one way to approach a math
problem, the way they are taught in school. It is important to
promote an environment in the math classroom that will en-
courage all children to solve problems in a variety of ways and
to encourage them to come up with their own unique strategies
(Franke, 2003). In the next section on problem solving we will
see that kindergarten children and first grade children can de-
velop a variety of invented and novel strategies when solving
challenging word problems (Carpenter, Ansell, Franke, Fen-
nema, & Weisbeck, 1993; Steinberg, 1985a; Warfield & Yttri,
Tsamir, Tirosh, Tabach & Levenson (2010) showed that it is
possible to enhance the development of flexibility and fluency
among kindergarten children. They gave 5 - 6-year-old kinder-
garten children a task that can be solved by multiple solutions
and solution processes. They gave the children 5 objects in one
group and 3 objects in the second group and asked them to
make equal groups without adding objects. They gave the task
to two groups of kindergarten children from a town with low
socio-economic status. The kindergarten teachers of one group
participated in a professional development workshop that en-
couraged them to engage the children in mathematically en-
riched environments. Project children produced more outcomes
and employed more methods than the non-project children. The
kindergarten children were flexible enough to employ more
than one method. The authors conjectured and brought evi-
dence from studies that the young students, who have had little
experience, may be more open and creative in their thinking
than older children who got used to standard ways of solving
problems in school when one solution and one way of solving
Problem solving in kindergarten and school can be a power-
ful tool to enhance learning with understanding and to develop
creativity in children. The ability to solve problems is very
important as a goal in itself. Young children aged 5 - 8 are able
to solve a variety of challenging problems if they have oppor-
tunities to solve problems on a regular basis (Carpenter, Fen-
nema, Franke, Levi, & Empson, 1999; Carpenter & Moser, 1984;
Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996).
Children who haven’t been taught to solve the problems suc-
ceed in doing so by applying their skills and understanding in
the new situation. Even kindergarten children can solve a vari-
ety of challenging word problems with different structures of
addition, subtraction, multiplication and division when they are
experienced with problem solving (Carpenter, Ansell, Franke,
Fennema, & Weisbeck, 1993; Warfield, & Yttri, 1999). The
young children solve problems with a variety of solution strate-
gies that are very innovative and usually are invented by the
children. They use their understanding of natural language,
showing sensitivity to the structure of the problem. They are
able to find correct solution strategies to different problems.
Their solution strategies have different levels of abstraction and
sophistication (Carpenter & Moser, 1984; Carpenter et al., 1999).
At the first level the child uses concrete objects to model the
math problems. The strategies at this level are called “direct
modeling” strategies. The child models the problem exactly by
its’ structure and does not transform one problem to another.
For example, to solve an addition problem with a missing ad-
dend such as: Danny has 7 cubes. How many more cubes
does he need so he will have 13 cubes? The child solves the
problem by adding on objects and not by subtraction. Young
children can apply the use of modeling with objects to different
problems and they find different correct “direct modeling” stra-
tegies. Examples of such strategies on different problems are
given in the results section for the problems Danny was asked
in the interview.
The second level of strategies is called “counting” strate-
gies (Carpenter & Moser, 1984; Carpenter et al., 1999). In these
strategies the child uses “double counting”—he counts two
series simultaneously. For example, to solve 7 + 4, the child
says 7 and counts (usually with fingers) 8, 9, 10, 11. The child
stops the counting when he shows 4 fingers. The fingers help
him see how many steps were used. There are a variety of
counting strategies for different problems. The third level of
strategies is called “mental strategies”—they are based on me-
morized facts. Either the child remembers the needed fact or
uses known facts to find unknown facts. These strategies are
called “derived facts” (Steinberg, 1985a). Example: 6 + 7 is 6 +
6 which is 12 and 1 more.
Young children usually just start to count at age 4 and some
of them may count up to 10. They begin to understand the
meaning of the counting (Gelman & Gallistel, 1978) and they
might add very small numbers in a context of a story. We will
Copyright © 2013 SciRes. 25
see that the child in this study is very far from a typical 4-year-
A number of studies have analyzed the structure of addition
and subtraction word problems and their level of difficulty
(Riley, Greeno, & Heller, 1983; Nesher, Greeno, & Riley, 1982).
Dynamic problems that have a verb to describe change are eas-
ier than static problems, especially in subtraction. In dynamic
problems, the place of the unknown in the number sentence
influences the difficulty of the problem. The hardest problems
are those with the unknown at the beginning. Carpenter et al.
(1999) related the difficulty of the problem to the child’s strat-
egy. A missing number at the beginning, for example, is diffi-
cult for a child who solves with direct modeling, since he does-
n’t know what number to start with. More details and explana-
tions are given in the results section in regard to the specific
problems that have been used in the interview.
Classes that promote problem solving are those in which the
teacher encourages children to solve challenging problems in a
variety of ways and explain and discuss their strategies (Fen-
nema, 1996; Franke, 2003; Steinberg, Empson, & Carpenter,
2004). The teacher builds on the children’s thinking in instruc-
tion. Socio-mathematical norms are developed in these classes
that give children a safe place to take risks and solve problems
in their unique ways (Yackel & Cobb, 1996). These classes are
very different than many familiar classes in which the teacher
tells the children how to solve problems and they drill the algo-
rithm. To develop creativity in children, it is important to start
with environments that promote problem solving at an early
Methods and Procedures
This is a case study with qualitative-descriptive analysis. I
conducted an interview with a very bright 4-year-old child to
see how he solves mathematical tasks. The child is very ad-
vanced for his age in his interest and knowledge of mathematics.
The interview was dynamic—the questions and problems I
gave the child to solve were decided upon as the interview pro-
gressed and as I could see what the child was able to do. There
were a few mathematical tasks that the child invented for him-
self and I went along with them and documented that as well.
The interview lasted about 40 minutes and was conducted at the
child’s home. The interview was videotaped and field notes
were taken. The videotape was transcribed and I watched it a
number of times so that I could see and notice more details. The
child, whom I call Danny (a pseudonym), was focused on the
mathematical tasks and was interested in them throughout the
interview. (His ability to concentrate on the math tasks for so
long is unusual for such a young boy and that allowed me to
give him many tasks). The math tasks were in 3 main topics:
counting and organizing objects and even and odd numbers,
solving word problems and working with large numbers and
place value ideas. There were 8 word problems with a variety
of addition and subtraction structures of the problems (with
numbers that cross to the second ten) and a multiplication
problem. The eighth problem was an addition with 2-digit num-
bers. The problems will be presented and analyzed in the results
In the results section I will present the mathematical tasks
and problems that I presented to the child. I will analyze them
to show their level of difficulty, the expected “direct modeling”
strategy that young children usually use to solve them and the
solution strategies that Danny used. I will also describe in-
vented tasks that Danny gave himself. I will add interpretations
to the data description.
Inventing Tasks for Himself: “I Wonder How Many
Are from Ea ch Color? What If I Add Them?”
Counting (with objects): I asked Danny: Can you count
from 12? He said he counts from 1. Can you count from 28?
He didn’t answer. What number comes after 28? He answers:
29. And what comes after 29? He answered 30. I took out
color plastic chips. He immediately sorted them by color. He
started to count the chips in each color group. He counted 10
pink ones and 8 yellow ones. He immediately said: “together
they are 18 because 10 and 8 is 18.” It seems that he remembers
the number fact and maybe has a good understanding of adding
a number to 10. He organized 9 blue chips in a line. I asked:
How many are the 8 and the 9 together? He immediately
answered 17. It appears that he has memorized the number fact.
In the next line he counted 7 and dealt with adding 9 and 7. He
didn’t recall it from memory and instead counted all the chips
and got 16.
Interpretation: Danny is a very curious young boy. He
makes up questions and tasks for himself. He decided to organ-
ize the chips by color, to count the number in each group and to
add two groups at a time. We will see that this curiosity is
characteristic of him throughout the interview. He makes up
problems for himself—some of them are very challenging. He
was very interested in the activity and didn’t wait for me to tell
him what to do. We can see that Danny already memorizes
addition number facts. He seems to understand adding a num-
ber to 10 and shows place value ideas. He can tell the next
number after 28 and after 29. He can go to the next ten easily.
He counts objects well most of the time.
More Explorations Danny Creates for Himself—“Can
I Organize the Number in Pairs?”
Danny continued to invent tasks for himself with a great cu-
riosity for explorations in doing mathematics. He spontane-
ously started organizing the color groups of chips in pairs. He
said—“can I organize 9 chips in pairs?” He made 4 pairs and
one. He did that for a few of the colored lines he made before. I
asked him if he knows how we call a number that we can put in
pairs? He said “2”. I told him we call it an even number and we
call a number odd if we cannot put it in pairs. I wanted to see
how quickly he learns something new and if he can use the new
knowledge. I asked him: Is 9 an even number? He answered:
“no, one is left over, There are 4 pairs and 1.” Is 6 an even
number? “Yes. There are 3 pairs.” Is 7 an even number? “No.
One is left over.”
Interpretation: Danny shows his curiosity and love to do
math here too by creating explorations for himself. He invents a
task—“can I organize the objects that represent different num-
bers in pairs.” He learns very quickly new ideas. He was able to
use the new terminology for even and odd numbers right away.
Solving Word Problems
Danny was asked to solve eight challenging word problems
Copyright © 2013 SciRes.
with different logical and linguistic mathematical structures.
The level of difficulty for the problems increased as he was
able to solve the previous problems. I will describe the charac-
teristics of each problem shortly together with a description of
Danny’s solution strategies.
Problem 1: Simple Dynamic Addition Problem with Sum
over 10: In the Kindergarten Backyard There Were 8
Children. Seven More Children Arrived. How Many
Children Are There Altogether?
The problem is a simple addition problem (8 + 7=). This is a
“dynamic problem” that has a verb, “7 children arrived”. The
sum of the numbers 8 and 7 is above 10. Danny remembered
the answer to the problem and answered 15 immediately with-
Problem 2: There Were 7 Children in the Backyard. How
Many More Children Need to Come So They Will Be 13
Children in the Back Yard?
The Problem: This is a “missing addend” addition problem.
It is dynamic (“How many children need to come?”) which
can help a young child understand more easily the “directing
instructions in the text” when he or she tries to model the prob-
lem with objects. The missing addend is in the middle (ask-
ing about the second addend, the number of children that need
to come). The number sentence that best describes the problem
is 7 + _ = 13. Most young children (first and second graders)
will not solve the problem by subtracting, but will model the
problem directly according to the addition structure of the
problem. Most children ages 5 - 8 use the concrete “direct
modeling” strategy to solve this problem (“adding on” strategy)
(Carpenter & Moser, 1984; Carpenter et al., 1999). They take 7
objects and then continue to add objects while counting from 7,
8, 9, 10, 11, 12, 13 and then count the number of objects added
and get 6.
Danny’s Solution Strategy: Danny didn’t know at first how
to approach the problem and it seems that he sees a problem
like that for the first time. He said “I don’t know”. He hesitated
for a short time and then he solved the problem by a “counting”
strategy (level 2). In this strategy he needs to coordinate “dou-
ble counting”—one count is the numbers from 8 to 13 and the
second count, that is done simultaneously, is the count of the
number of adding steps (1-6). Usually, children “double count”
by keeping track on their fingers. Danny also used his fingers
but in a different way. He didn’t count the numbers in sequence.
Instead, he kept track of the “double counting” by saying one
number from one sequence (8-13) and one number from the
second sequence (1-6). He put one finger and said “7 plus 1 is
8”. He put 2 fingers and said “8 plus 1 is 9” and so on until he
got to 6 fingers and the sum 13. He had some difficulty finish-
ing in this way and I helped him and we did it together until he
got to the answer 6 (6 fingers).
Interpretation: Danny used a correct “counting” strategy
and dealt with the double counting so he can keep track and
knows when to stop the counting. His keeping track process is
hard to execute since he doesn’t say the numbers in sequence 8,
9, 10 ··· but he takes one number from one counting sequence
and another number from the second sequence. He needs to
remember where he was in each sequence when he comes back
to it, which puts a strain on the short term memory. This is a
relatively rare keeping track strategy (Steinberg, 1985) and
children who use it often get confused in the middle and make
errors, since it is so difficult to do. Even though he is using a
level 2 “counting” strategy and he does not use a concrete “di-
rect modeling” strategy, still, the structure of the missing ad-
dend problem influences his solution. He solves the problem by
addition and doesn’t transform the problem into a subtraction
situation. He tries to follow the story in the problem exactly.
Problem 3: A Multiplication Problem: There Were 3
Groups of Children in the Kindergarten. There Were 4
Children in Each Group. How Many Children Were in All
This is a multiplication problem. It is reasonable to assume
that Danny has never met a problem like that before. Young
children usually solve the problem by directly modeling it with
objects or drawing. They make 3 groups, put 4 objects in each
one and count them all. Danny didn’t use objects. He answered
immediately by recalling the multiplication fact from memory
(level 3—recall of facts). He said: “3 multiplies ··· that’s 12.” I
asked what number sentence was he saying and he didn’t an-
swer. As he explained his strategy he had solved it in a dif-
ferent way, checking his solution: He said: “4 × 2 = 8 and 4
more is 12. He found that by a “counting strategy” (level 2)
similarly to how he solved the previous problem with keeping
track strategy of one from each sequence. He put up one finger
and said: “8 plus 1 is 9”, a second finger and said “8 plus 2 is
10”, a third finger and said: “8 plus 3 is 11”, and a fourth finger
and said: “8 plus 4 is 12”. I asked him how he knew this is a
multiplication problem—he didn’t answer.
Interpretation: He understands a multiplication problem.
He recognizes that this is a multiplication situation and he
knows how to say a multiplication number sentence (even
though he said it without finishing his thinking). He knows this
multiplication fact from memory. He checks his answer with
another solution strategy. He shows flexibility and fluency by
coming up with two ways of solving the problem spontaneously.
He is able to change his point of view on the problem and to
come up with a different solution strategy.
Problem 4: Subtract ion, Star t Unknown: There Were a Few
Children in the Kindergarten Backyard. We Do Not Know
How Many Children Were There. Then, 5 Children Went
Inside the Kindergarten and 4 Children We re Left in the
Backyard. How Many Children Were at the Backyard at
This is a dynamic subtraction problem (“5 children went in-
side”). A number sentence that represents the problem best is:
_ − 5 = 4. This is a very difficult problem for children who
solve problems in a “direct modeling” strategy (level 1) using
concrete objects. This problem asks to find out how many chil-
dren were present initially. Young children who solve the
problem using direct modeling strategies have great difficulty
modeling the problem, since the unknown is at the beginning.
The child doesn’t know what objects to start with and how to
represent the unknown. At the “direct modeling” level the chil-
dren are not able to see the relationship between this subtraction
problem and an equivalent addition problem, and they do not
transform the problem into addition. Many first and second
graders are not able to solve the problem at all. The children
who solve the problem by “direct modeling” use a “trial and
error” strategy. They guess a number to start with (such as 10),
Copyright © 2013 SciRes. 27
they subtract 5 from 10 and see if 4 is left. If not, they adjust
their guess and try to start with another number. It is unusual to
give this problem to young children, definitely at age 4.
Danny’s Solution Strategy and Interpre tation: Danny first
answered he doesn’t know. He was embarrassed. He asked “1?”
I suggested for him to try. I repeated the problem. He put down
one object and said “it is not 1”. He put 2 objects and said—no,
it is not that 3?—No. 4?? I repeated the problem. I asked if
there were 4 children, is it possible that 5 children had left? He
said—no. That seems to help him understand the problem. He
continued to add one at a time and checked to see what is left
after subtracting 5. Interestingly, he didn’t use the objects to do
the subtraction. He did all calculations by recall of facts from
memory. But he needed to put down the objects so he can think
of the number he subtracts from. After he understood the prob-
lem, he used a “trial and error” strategy. This strategy is con-
sistent with the structure of the problem—a subtraction problem
in which the start is unknown.
Problem 5: A Compare Problem: Danny Has 11 Cubes. His
Brother, David has 7 Cubes. How Many M ore Cubes Doe s
Danny Have than David?
This is also a difficult problem even for second graders. The
idea of comparing two sets and the language involved in it are
not easy for children (Carpenter et al., 1999; Riley et al., 1983).
There is no verb in the problem (it is static) that could give the
child a hint towards how to act out the problem. Many children
who use “direct modeling” have difficulty solving the problem,
others build the two groups and match them one by one and see
how many more counters are in the larger group. Danny first
was not sure what the problem is asking. I probed him by:
“does he have 1 more? 2 more?” He understood the meaning
and solved it with a “derived fact” strategy based on knowing
addition facts from memory (level 3). He said: “7 and 3 is 10.
So 7 and 4 is 11”. He used a known fact (7 plus 3) to solve an
unfamiliar fact—7 plus 4.
Problem 6: Subtraction Missing Addend P roblem: Danny
Had 15 Cubes. His Brother, David Took a Few Cubes and 8
Cubes Were Left. How Many Cubes Did David Take?
This is a subtraction, dynamic problem. The number sentence
that represents the problem best is 15 − _ = 8. A child who uses
“direct modeling” takes 15 objects, starts by trial and error to
take some counters away and checks if 8 counters are left. Then
the child corrects the trials until 8 counters are left. This strat-
egy models the problem by its structure. The child who uses
level 1 “direct modeling” strategy doesn’t see the relationship
between this subtraction problem and an addition problem.
Danny did see the connection between addition and subtraction
and he transformed the problem into a missing addend addition
problem. He said: “8 plus what is 15?” It is easier to do the
calculation in addition. He solved it with a “counting strategy”
with double counting simultaneously, similarly to previous
problems: He put one finger and said “8 plus 1 is 9”, 2 fin-
gers—“8 plus 2 is 10” all the way to “8 plus 7 is 15” and gave
the correct answer 7. This keeping track process is very diffi-
cult to execute especially with 7 steps (Steinberg, 1985b). Here
too, Danny doesn’t count continuously in each sequence such
as 9, 10, 11··· but needs to go back and forth between the 2
counting sequences (9-15, 1-7) and needs to remember where
he was in one sequence and where he was in the second se-
Problem 7: A Static Subtraction P roblem: There Were 12
Cubes. 4 Cubes Were Blue and the Rest of the Cubes Were
Yellow. How Many Yellow Cubes Were There?
This is a difficult problem since it is static. There is no verb
or change in the problem.
A child who solves the problem by “direct modeling” (level
1) doesn’t get cues from the problem of how to act it out with
objects. Danny solved the problem immediately in the most
surprising and creative way. He understood he needs to subtract
4 from 12. He connected the subtraction problem to a multipli-
cation problem. He remembers the multiplication fact:
3 × 4 = 12. Therefore, 12 − 4 = 8. He said: “If 3 times 4 is 12, I
take one 4, so two 4’s are left and this is 8 (he remembers how
much are two 4’s)!!
Interpretation: In the many years that I have interviewed
school and kindergarten children on their mathematical think-
ing I have never seen a child who solved a subtraction problem
via a multiplication problem. This is a very original and crea-
tive solution. The originality of it is very unique and rare. Of
course, it is very unusual that this bright 4-year-old child re-
members multiplication facts at this age. Danny understood the
problem right away and did not find the static situation difficult.
He recognized that he can represent the problem with a subtrac-
tion number sentence. He solved it with “mental strategies”
(level 3). This strategy is based on memorized number facts. He
used “derived facts” in which he used a known multiplication
fact 3 × 4 to find an unknown subtraction fact 12 − 4. This so-
lution is also very flexible. He saw the relationship between
multiplication and addition and subtraction. 12 − 4 is 3 times 4
minus one four. Two 4’s are left and he knows it is 8 by an
addition or multiplication fact, 4 and 4 is 8 (or 2 times 4 is 8).
Working with Large Numbers and Place Value
I wanted to see Danny’s knowledge of large numbers and
place value ideas. I asked him how many children are in his
kindergarten. He said 23. And how many are in the second
kindergarten? He said 31. I dared asking him how many chil-
dren are in both kindergartens. He said he doesn’t know and he
had no idea how to calculate the answer. I showed him base-ten
manipulatives (from cardboard) that I brought to the interview.
These are strips divided into 10 squares for tens and small
squares for ones. I asked him to count how many squares are in
one strip. He counted 10. I showed him 2 strips and asked how
many squares are there? He immediately said 20. I added a one
and after a short hesitation he said it is 21! I added another 2
and we counted them together: 22, 23. I told him this shows
how many children are in his kindergarten. I asked him to show
how many children are in the second class. He took a 10 and
said 10. He took another 10. I asked how many are there and he
said 20. I asked how many are 20 and another 10 while giving
him the 10 and he said 30. And one more? He said 31. Now I
asked how many children are there in both kindergartens? He
was not sure what to do and said he doesn’t know. I suggested
he can count. He looked at it and immediately said 50 pointing
to the tens. Then he proceeded to calculate orally without using
the 10-blocks. He said 3 and 1 is 4, so it is 54!! (Figure 1).
I asked him if he knows how to write 54 and he wrote it well.
I dared asking another problem with the big numbers. I asked:
“How many children need to come so we will have 60 chil-
dren?” He calculated mentally without the blocks. He cleverly
Copyright © 2013 SciRes.
said 54 plus 5 is 59, and 1 more is 60. He wrote a 6 under the
54 and wrote 60 under (Figure 2).
Interpretation: Danny has a lot of knowledge of large
numbers and place value ideas. It was remarkable to see how
fast he learned to use the new 10-blocks to solve very chal-
lenging problems with large numbers. He can count by tens. He
didn’t need to count the second and third ten by ones, but could
see it is 20 and 30 immediately. He could look at 3 10-blocks
and 2 10-blocks and know it is 50 without even counting by
10’s. He seems to grasp the idea already that 5 rods of 10 is 50.
He could count on from tens—20, 21, 22, 23. He could add tens
and ones separately. To get from 54 to 60 he first added within
the same ten—and he could do it mentally—54 and 5 is 59.
Then he saw that just adding one more is 60. It appears that he
is familiar with the sequence of the numbers at least through
The numbers are meaningful for him and he uses them for
general knowledge from his surroundings. I assume not too
many 4-year-old know how many children are in their class and
in the parallel class.
Reading and Calculating with Large Numbers: I asked
him to read the following numbers: 124, 1335, 2034—he could
read all of them easily. I asked how much is 2034 + 4. He an-
swered immediately 2038!!
Inventing Another Problem for Himself: I stopped the in-
terview at this point thinking to myself that the child was so
Using tens and ones manipulatives to solve 23 + 31.
Danny writes large numbers.
concentrated on solving these amazingly difficult problems—he
is only 4! But it was not enough for Danny. He came up with
another invented task for himself. He saw that there are more
manipulatives in my kit (a hundred—a square and a thousand—
a drawing of a cube). To my amazement Danny pointed at the
thousand cube and said—this is a thousand. I asked “How do
you know”? And he said it looks “a lot” and it is probably a
1000!! He asked how many are in the square and I said a hun-
dred and I showed him that 10 tens equal a hundred. He created
a task for himself—he said, “I wonder how much is a 1000
(taking the cube) plus a hundred (taking the square) and a ten
(taking the 10-rod) and a one (taking the one)” (Figure 3). He
calculated the numbers mentally. He said: “1 and a 10 is 11. 11
and a 100 is 111. And another 1000? He didn’t know how to
proceed. I suggested he starts from the 1000 and 100. He could
say it is 1100. And then he added the 11 and said 1111. His
mom asked him how to write it. First he said 1011 and when
she asked him again he corrected himself to 1111. His mom
told him what a unique number it is that all the digits are 1.
Interpretation: Here again Danny shows how quickly he
learns new ideas. He has a very good understanding of the struc-
ture of the number system. He can read large numbers. He can
add a number in the same ten with a large number (54 + 5 = 59),
can add hundreds and tens (11 + 100 = 111) and thousands
(2034 + 4 = 2038, 1000 + 100 + 10 + 1). He can write 2-digit
Wow. I am totally “blown away”. All that was done by a 4-
year-old child?!! In one sitting!
This study shows that it is possible for a 4-year-old child to
do mathematics at a very high level. Danny is able to use
mathematical ideas flexibly, with much insight and with a good
number sense. He could count and remembers many addition
and multiplication facts. When he didn’t remember he used
“counting” strategies. He has much knowledge of large num-
bers and place value ideas. He is curious and intrigued by
numbers and mathematics and has high motivation. He also
learned new ideas during the interview very quickly and was
able to use them. Thus, we can see some information on his
learning ability while interacting with an adult, what Vygotsky
(1978) called “the zone of proximal development”.
Problem Solving Ability: Danny solved all 8 word problems
with different structures. The problems are very challenging
Showing 1000 + 100 + 10 + 1 with manipulatives.
Copyright © 2013 SciRes. 29
and most of them he saw for the first time. He was attentive to
the structure of the problem and to the nature of the language in
it. Danny didn’t use “direct modeling” (level 1) strategies with
concrete objects (which most 5 - 7-year-old children use). He
solved the problems with recall of memorized number facts
(level 3) or with “counting” strategies (level 2). Still, he was
listening carefully to the structure of the problem and typically
used strategies that matched the problem. For example, he
solved the subtraction problem with the “start unknown” (prob-
lem 4) with a trial and error strategy in which he guessed a
number to start with and then adjusted it after a check—since
he needed a number to start the subtraction process. Modeling
the problem according to its structure (even when not using
“direct modeling”) is a very powerful tool that enables even
kindergarten children to solve challenging problems (Carpenter
et al., 1993; Warfield & Yttri, 1999).
Learning to solve challenging word problems should be a
major goal of teaching mathematics in kindergarten through
elementary school (Carpenter et al., 1999). It is more impor-
tant than learning to add or to multiply. The children learn to
use their mathematical knowledge and skills in new situations
in creative ways. This kind of adaptive use of one’s thinking is
needed in real life and in changing situations. By encouraging
kindergarten and school children to be engaged in solving
problems and finding their unique solution strategies we en-
hance creativity in all children (Fennema et al., 1996; Silver,
1997). Even children who solve problems only with “direct
modeling” strategies with objects come up with many different
and correct invented strategies according to the structure of the
problem. Children see that solving a challenging problem can
take time, they learn to deal with an uncertain situation and to
develop perseverance so they can stick with the difficult prob-
lem even when they do not see an immediate solution. This is
very important for developing a sense of self confidence in
one’s ability to solve problems.
Personal Traits That Are Important for Problem Solving
Ability: Danny has great interest and motivation to do math
(Figure 4). He is also willing to “take risks” and to try to solve
new and challenging problems, even when he has never seen
problems like that before and is not sure he can solve them. He
uses the skills and ideas he already has to tackle the new and
unfamiliar problems. Leikin & Pitta-Pantazi (2013) summa-
rized studies of personality attributes of talented children and
found that risk taking to tackle an unfamiliar problem is char-
acteristic of creative children and is one of the traits that pro-
mote innovation. Danny also learns fast—there were a few
Danny shows interest and motivation to do math.
times during the interview that he learned new ideas and skills
and was able to use them immediately. Examples: He learned
how to use tens and ones with 10-blocks and learned to add
2-digit numbers with them. He learned about an even number
and could immediately check if numbers are even or odd.
Danny was also very concentrated on the tasks throughout the
interview—it is very unusual for a 4-year-old child.
Creativity: Danny showed a lot of creativity in doing mathe-
matics. Examining Torrance’s categories for creativity (1974):
Flexibility, fluency and originality in Danny’s work we can
see many instances in which he shows creativity. Flexibility:
While Danny usually solved problems according to their stated
structure, he was able to see connections between some prob-
lems and to transform one problem into another problem that
was easier for him to solve. Thus, he shows flexibility in these
solutions. In Problem 6 the given structure was 15 − _ = 8 but
Danny solved by 8 + _ = 15. This requires great flexibility in
thinking and seeing the relationship between addition and sub-
traction. Similarly, he solved 12 − 4 in problem 7 by 3 × 4 = 12
and one 4 less is 2 × 4. Solving the subtraction problem by a
multiplication problem shows great flexibility. Flexibility in
solving word problems develops with age. Young children
(ages 5 - 7) are usually not flexible in their solution strategies
and they model problems exactly by their structure (Fennema et
al., 1996). Another example where Danny showed flexibility
was when he tried to add 111 + 1000 and didn’t know how to
do it. When I suggested he can start with the 1000, he was able
to switch and solve the problem. Fluency: Danny spontane-
ously solved a problem in two different ways. For example, in
problem 3 he knew that 3 × 4 is 12 and he checked it by 2 × 4 is
8 and 4 more by counting. He played with the numbers and
looked for patterns. When he organized objects in color groups
and counted and added them, he was able to find another task
with the same objects—to organize them in pairs. This playful
attitude helped him change view on the same situation and it
develops fluency. Originality: Even though Danny is young,
he showed originality in doing math. His solution of solving 12
− 4 by multiplication is very unique and original. He needed a
great deal of mathematical knowledge for that—he had already
memorized multiplication facts. Inventing tasks: He invented
and explored many mathematical tasks for himself. This shows
originality. Examples: “I wonder how many objects are from
each color? How many are they together? Can I put them in
pairs? I wonder how much is 1000 + 100 + 10 + 1?” By in-
venting explorations for himself, Danny learns a lot. Many of
the situations are very original and unique. Few kindergarten
teachers would dare to give him such problems. I recently
talked to Danny’s mother and she told me he spent hours with a
task he invented—to find all multiples of 3 up to 1200!!!
What Kind of Support Does Danny Ge t at Home? When I
see what Danny is able to do at age 4, l am amazed. I wanted to
know what kind of support he gets at home. How does he de-
velop his creativity and meaningful sense making at such an
early age. His mother is the main person that helps with his
math learning; I asked her how he learns mathematics and what
kind of interactions or opportunities he has at home. The
mother answered that Danny initiates a lot of the activities and
tasks by himself and she responds to that and tries to help him
pursue his interest. She tries to respond in a way that will allow
him to think and to find the answer himself. She usually an-
swers his question with another question. She said: “I was
charmed by his world and learned to see things through his very
Copyright © 2013 SciRes.
mathematical eyes. He is very attentive to his surrounding and
asks many questions”. Danny showed interest in numbers from
a very early age (2). When they were taking an elevator he was
attentive to the numbers so they would go up and down the
elevator many times so he can see the changing numbers. One
time they were in a building with a −1 floor. He was very ex-
cited by that and they again went up and down so he can see
that −1 comes when they go below 0. When Danny goes in a
car he sees excitedly that the stoplights are numbered and
spends much time reading and thinking about these numbers.
When his grandparents take him to the zoo, he counts the num-
ber of animals in each cage and compares them. We can see
that Danny’s great curiosity about math is enhanced by the
interaction with his mom and family. When I came for the in-
terview there were numbers (including large numbers) hanging
on the walls. They count leaves of a flower and how many
leaves are left after some fall. They divide a cookie and talk
about halves. Danny learned to write the number “one million”.
He asked his mom and she checked for him how you call a
number that has 60 zeroes. Besides his explorations, Danny
also gets a lot of drill and practice. He uses games and practice
on electronic devices and practices addition, subtraction and
multiplication facts as well. So we can see that, in spite of
Danny’s very early age, he spends many hours learning, explor-
ing and practicing mathematics.
In examining the support that Danny receives from his
mother, we can see that she views her role as helping Danny
with his ideas, his interests and his ways of solving problems.
The mother-child interactions are within a framework of look-
ing at mathematics as a broader tool to explore real life situa-
tions and to be engaged in problem solving. Studies show that
there are different patterns of parent-child interactions at home
when providing assistance in relation to the child’s learning of
mathematics. On the one hand the parent interaction with the
child was “school like”—structured, directing the child to one
kind of solution process and answer and controlling the activity.
Tiedemann and Brandt (2010) describe such a parent who “ex-
pects a specific answer and constricts the possible course of
actions for the learner until the latter can give the requested
answer”. Danny’s mother appears to perceive her role in help-
ing Danny with his math learning at the other end of the spec-
trum as described by a few researchers (Bishop, 2002; Tiede-
mann & Brandt, 2010; Sfard & Lavie, 2005). She was respon-
sive to Danny’s thinking and acts, and encouraged him to ex-
plore, to come up with his own ideas and to perceive learning
mathematics as part of solving problems within broader life
situations. Leder (1992) found that such an approach is impor-
tant for the child’s development, especially when the parent
poses high cognitive level questions and when he or she en-
courages the child to be autonomous.
Implications for Education: Kindergartens and schools can
play an important role in enhancing all children’s ability to
learn mathematics meaningfully and with confidence. They can
encourage children to be involved with challenging problem
solving. Teachers can expect children to solve problems in their
own unique solution strategies, to reflect and discuss the solu-
tions and the mathematical ideas and to learn to respect each
other’s thinking. Teachers can build upon the students’ work in
instruction. Research reviewed in this article shows that con-
structivist classes like that develop understanding and creativity
among the children (Fennema et al., 1996; Steinberg et al.,
2004). Children invent many solution strategies and procedures
and are exposed to a variety of solutions (Franke, 2003). This
helps them develop flexibility and fluency and more children
also suggest original strategies (Silver, 1997). Teachers are sur-
prised to see the richness of the ideas all children bring to class.
Class discussions are also very interesting when they build
upon children’s thinking (Steinberg, Empson, & Carpenter,
2004). In these classes, students who are very strong in learning
mathematics also find challenging problem solving tasks and
high level of mathematical ideas and the environment encour-
aging them to be creative. In these classes children accept the
socio-mathematical norms (Yackel & Cobb, 1996) that solving
in a variety of ways is allowed and encouraged.
These classes are very different from most classes we are
familiar with all over the world, in which “tell and practice” is
the main way of teaching—the teacher or the textbook tells
children exactly how to solve and then they practice it. There is
usually one way the students are expected to solve math prob-
lems and they are not encouraged to deviate from the standard
way of doing things. These classes do not encourage problem
solving and creativity.
In this paper we had the opportunity to “meet” a very special
4-year-old child who showed us that the “sky is the limit”. We
can hope that he finds himself in a challenging and motivating
classroom that will encourage problem solving and creativity.
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