Children are Born Mathematicians: Encouraging and Promoting Early
Mathematical Concepts in Children Under Five

Dr. Eugene Geist

Ohio University

108 Tupper Hall

Athens Ohio 45701

Children, from the day that they are born, are mathematicians.  They are constructing knowledge constantly as they interact mentally, physically, and socially with their environment and with others.  Young children may not be able to add or subtract, but the relationships that they are making and their interaction with a stimulating environment promotes them to construct a foundation and framework for what will eventually be mathematical concepts.   There is even some evidence that some mathematical concepts may be innate.

            Perhaps it is time that we begin to look at the construction of mathematical concepts the same way that we look at literacy development - as emergent.  The idea that literacy learning begins the day that children are born is widely accepted in early childhood.  Children learn language by listening, and, eventually, speaking and writing language and this language learning is aided by the innate "language acquisition device" which acts as a blueprint for grammatical development and language learning. Reading to infant, toddlers and preschoolers is known to be an early predictor of positive literacy because these activities promote and support learning to read and write by immersing children in language and giving them an opportunity to interact with it.

I propose that just as Chomsky has shown strong evidence for an innate "language acquisition device" that provides humans with a framework for learning language, there is a "mathematics acquisition device" that provides a framework for mathematical concepts.

            If such a device were present, we would expect children to

1) Naturally acquire mathematical concepts without direct teaching,

2) Follow a generally standard sequence of gradual development, and

3) Most importantly, we would expect to see evidence of construction of mathematical concepts from a very early age. 

Let me take these points in order and try to offer evidence for them.

1) Naturally acquire mathematical concepts without direct teaching,

            With close examination of young children and especially infants and toddlers, we see that many of the foundations for mathematics are never directly taught to children.  In fact, I challenge anyone to propose a feasible way to actually achieve this task with children 4 and under.  No, the way that these children learn these concepts is through construction and interaction with their environment.  Teachers may help by setting up an interesting and stimulating environment; the child's mind is actively making all kinds of relationships and organizing those into concepts that will become mathematics.

            The child's mind seems to know what to do and all normal children seem to have no difficulty constructing concepts of number, order seriation, or classification, well in advance of any teaching. Children begin to construct the foundations for future mathematical concepts during the first few months of life.  Before a child can add or even count, they must construct ideas about mathematics that cannot be directly taught.  Ideas that will support formal mathematics later in life such as order and sequence, seriation, comparisons, and classifying all are beginning to emerge as early as infancy. 

The seemingly simple understanding that numbers have a quantity attached to them is actually a complex relationship that children must construct.  This concept is the basis for formal mathematics and it is a synthesis of order, which is the basic understanding that objects are counted in a specific sequence and each object is counted only once; seriation, which is the ability to place an object or group of objects in a logical series based on a property of the object or objects; and classification, which is the ability to group like objects in sets by a specific characteristic.  This synthesis takes place by children interacting with objects and putting them in many different types of relationships.

Children even younger can be seen to use their developing understanding of order, seriation, classification, and natural problem solving ability.  I observed an 18-month-old child playing in a large pit filled with different colored balls.  The child dropped over the side of the pit one ball, then a second ball, and then a third ball.  The child then went to the opposite side of the pit and dropped two balls.  He then went back to the first side, reexamined the grouping of balls, moved to the second side and dropped another ball over the side to make that a grouping of three.

            This may seem unimpressive by adult standards, but for an 18-month-old child, the coordination and comparison of threes on opposite sides of a structure is evidence of this child making a mathematical relationship.  It is not yet a numerical relationship because the child is using visual perception to make the judgment of "same" or "different".  However, the coordination of dropping three balls each time is evidence of an understanding of  "more" and "less" and basic equality.  The child may not be developmentally ready for counting and quantification, but this simple task shows that children as young as 18 months can make some rudimentary mathematical relationships.  Teachers of infants and toddlers need to be aware of these actions and abilities and help provide activities to encourage construction of these mathematical concepts.  Activities that promote children to make many different relationships between and among objects, to interact with other children and adults, and to mentally and physically act on objects promote this type of construction.

            Although these basic mathematical concepts cannot and should not be directly taught, educators of young children need to emphasize and encourage children's interaction with their environment as a means of promoting and encouraging emergent math concepts.  Children's logic and mathematical thinking develop by being exercised and stimulated.  Teachers who promote children to put objects into all kinds of relationships are also promoting children's emergent understanding of mathematics.

            Making sure that children from birth through age four have a stimulating environment and opportunities to make many different kinds of relationships as early as the first months of life can support the child's emerging understanding of mathematics.  Teachers in infant, toddler, and preschool programs can do a number of things, like offering objects to compare, using beat and music, modeling mathematical behavior, and incorporating math into every day activities, to facilitate the emergent mathematician within every child.  The basic frameworks for math cannot be directly taught but can be easily promoted in the classroom.

2) Follow a generally standard sequence of gradual development, and

            As with a lot of development and developmental theories, we would expect a natural developmental pattern for things that are developmentally brain based rather than the product of internalization from outside teaching.  This is exactly what we see in mathematics.  As a matter of fact, we see similar relationships in math to the way that language develops.  So do we see a natural sequence unfolding in mathematics?  Yes we do.

            An example of this is an interaction I had with a 3 year old.  Her parents had asked her to say her numbers for me and she correctly counted to 20 with no errors.  I then pulled out 20 pennies that I just happened to have in my pocket.  I asked her if she could figure out a way to make sure we both got the same number of pennies.   She looked at the pile of pennies, split the pile down the middle, and slid a handful over to me and she took the rest.  My pile contained 12 pennies and hers contained 8.  I asked her how she knew we had the same amount and she attempted to count the pennies by pointing at the pile and saying "one, two, three, four, five, six, seven."  However, she did not have an understanding of the importance of order, and therefore, counted some pennies twice and missed some completely. I then asked her to count hers and she counted ten.  When I asked her again if we had the same amount, she made another quick visual inventory and replied "yes."  I then lined up eight pennies in a row and asked her to make a row with as many pennies as I had lain out.  She took the rest of the pennies (12) and made a row below mine.  I again asked her if there were the same number of pennies in each row. She counted her row and replied "yes, see, One, two, three, four, five, six, seven, eight, nine, ten."  I asked her to count mine and she came up with eight.  I asked her again if they had the same number and she again replied "Yes".

            This is a good example of a child who is not yet able to coordinate order, classification, and seriation and therefore, cannot put the pennies in a "quantity" relationship.  Children as young as two may be able to count to 10 or even 20, but if they do not link their counting to quantification it is no different from memorizing their "ABC's" or a list of names like "Bob," "Joe," and "Sara."  This is why this child could not make a numerical relationship between the two sets of objects.

            She used visual cues to estimate the sameness and difference of the sets instead of using number.  Her logic and problem solving ability is still perceptually bound.  However, as she continually interacts with the objects and with other children and adults, she will come to realize the limits of her solution and begin to construct new ways of reaching a solution.  This type of confusion or what Piaget called "disequilibrium" (Piaget, 1969) is what leads the child to make further constructions and strengthen her understanding of mathematical concepts.

            Eight months later, I again had an opportunity to interact with this specific child.  We again played the game with the pennies.  This time when I asked her to divide up the pennies she used a one-to-one correspondence method.  She gave me a penny and then one to herself until all the pennies were distributed.  When I asked her how she knew we had the same number, she counted each penny in a specific order and only once to get the correct answer.

            I collected all the pennies back into one pile.  I then showed her one more penny and added it to the pile.  I asked her if she saw what I had done and she said, "Yes, you added one more penny!"  I then asked her to figure out a way to divide up the pennies and make sure we both got the same number.  She used the same method of one-to-one correspondence she had used previously.  I asked her if we had the same number of pennies and she replied, "Yes."  I asked her to count them and when it turned out that I had one more penny, she was quite perplexed.  She could not figure out how that had happened.

            The child has made significant progress in her understanding of basic mathematical concepts.  Her method of dividing up the pennies is no longer visual.  She is using number concepts to solve her problem.  However, her understanding of this mathematical concept is still weak and breaks down when strongly challenged.

3) Most importantly, we would expect to see evidence of construction of mathematical concepts from a very early age. 

And we do.  We see evidence of infants and toddlers sorting objects, stacking objects, and banging things together.  Not math you say?  Well maybe not as we think of it as adults, but remember, infants and toddlers are still constructing the concept of number and more basically the concept of "one".  Think about how you would teach an infant or toddler the concept of one.  I can't think of a way.  Yet eventually almost all children without a serious mental defect achieve this task (and many others), even ones that were not taught.

So if we think of mathematics as an innate ability and assume that children have a "Mathematics Acquisition Device" in their mind, this does not mean that we abandon teaching mathematics to children, but it does mean that we have to reexamine some of the ways we teach mathematics.

          Even though there is a common sequence and children have this device, all children are different.  They learn at different rates, they have different interests, they have different talents, they have different modes of learning, and when they come into the classroom they are all at different levels of understanding of mathematics. Curriculum for young children must be molded and customized to meet the needs of all children in a particular classroom.  It must be flexible and adaptable so the teacher can use their knowledge of the children's prior understanding to create a mathematics program that meets the child at that level, and stimulate construction of more complex mathematical understanding

If we think of mathematics as developmental and being helped out by a "Mathematics Acquisition Device", how would we see children interacting with mathematics?  Well we would see children doing math independently.  And we do.  Young children take delight in sorting and counting even when it is not part of a formal lesson.  They love games and puzzles where math is central.  The biggest thing that adults need to do to foster a love of mathematics is to stay out of the way.  Children develop math phobias and bad attitudes toward mathematics because of things that adults and teachers do, like high stakes testing and timed tests.

Concepts will develop without direct teaching.  Children by using their natural thinking ability and their proclivity for mathematics will naturally develop mathematical concepts.  This does not mean that the adult does not have a role, we do - an important one.  But that role is more as a facilitator than a teacher.

We see children using math to make sense out of their world.  It is accepted that mathematics is a universal language.  We even assume that alien species will have constructed the same mathematics as we have.  And just as physicist use mathematics to understand the universe, children use mathematics to understand their world.  Even infants understand the concept of "more".  It is one of the first math concepts they construct.  Even 6 month olds can let their parent or caregiver know they want more food or milk.

So if we are to change the way we think about mathematics and how it is taught to young children and if there were such a thing as a "Mathematics Acquisition Device", what changes in teaching mathematics to young children would we see.  Well to begin with we would begin to treat young children as young mathematicians.  Instead of sitting them in rows and having them memorize, we would try to have them invent or discover mathematical concepts and new mathematical ideas the same way that mathematicians solve more complex problems.  So what are some of these ways that real mathematicians work?

Mathematicians often work for a time on a single problem

          Mathematicians may spend months and years thinking and working on a proof to one problem.  Students, also, should be allowed ample time to work on one problem. To do this, students need to be offered fewer problems and more time to complete them enhancing their problem solving abilities.

Mathematicians collaborate with their colleagues and study the work of others

          Social interaction is one of the most important parts of being a mathematician.  A mathematics classroom, especially one that views students as young mathematicians, should include many opportunities for social interaction   Children are usually not encouraged to defend an solution or collaborate on solving a problem.  Instead they are given individual practice worksheets and asked to complete them quietly (Fosnot, 1989).

          If children are going to be viewed as young mathematicians, they must be allowed to collaborate, argue, consult, defend, ask, explain, and pose to, and with, other students using mathematical ideas.  Children construct mathematics understanding through this type of social interaction.  Without this interaction, children just memorize how to get a certain solution without developing understanding.

Mathematicians must prove that for themselves their solution is correct

          Mathematicians must question assumptions and understand the mathematics behind an answer.  Mathematicians must prove to themselves and others that their solution correct.  If students are taught merely to memorize answers and constantly rely on a teacher to tell them if they are correct or, then the important process of proving a solution is removed from students.

The problems mathematicians work on are complex

          Complex problems promote problem solving abilities.  Children, like mathematicians, should be immersed in complex problems that require mathematical problem solving and complex numerical thinking.  Good problems ask students to find innovative solutions to the problem without a time limit being set on their thinking process (Wakefield, 1997).  Problems can and should spark discussion and even disagreement among the children.

Mathematicians get satisfaction from the process

          Children will understand mathematical concepts and procedures more thoroughly if they are allowed to use their own thinking process to explore mathematics (Kamii, Lewis, & Jones, 1993).  It allows them to make connections to what they already know and to real life experiences.

          In the process of discussing and comparing different methods that children use to reach solutions children strengthen their understanding of both concepts and procedures.

Mathematicians have a sense of pride in getting a solution

          Children can get very excited about a mathematics problem and children find pleasure and excitement in problem solving (University of Chicago, 1998).  If children are allowed to think for themselves and discuss and defend their ideas, mathematics becomes just as fun as trying hard to complete a video game or working diligently to put a puzzle together.

          Mathematicians Use Unsuccessful Attempts as Stepping Stones to Solutions

            For children to be treated as mathematicians, they should realize that they may have to try many different approaches before they reach a solution.  Emphasis needs to be placed on the valuable mathematical thinking going on in the child's mind.  It should be emphasized and modeled to children that unsuccessful attempts and errors can be stepping-stones to solutions.

          Children have a natural curiosity and zeal for exploration and understanding that applies to learning mathematics.  If children are encouraged to act like young mathematicians and use their natural thinking ability to attack and solve problems, as we see in the classrooms of Japan, mathematics becomes not a chore but a challenge to the student (Wakefield, 1997).

          Excitement about mathematics should be the goal of every teacher of mathematics.  Children from early childhood on must be treated as if they were young mathematicians.   This philosophical change is not made by more emphasis on skill and drill methods or adding more mandatory tests (Kelly, 1999).  It will take a deliberate process of change in the way children are viewed and treated in classrooms (Bay, Reys, & Reys, 1999).        

We must treat children as mathematicians from the beginning.  We can foster the mathematical thinking skills required by offering materials and experiences that bill build a strong foundation for future mathematical learning.

So lets look at a few things teachers can do to help promote mathematics in young children.

Birth to Two

          Infants and toddlers are exploring their environment using their senses.  Piaget (1969) called this time the Sensory-Motor stage because they explore and learn about their environment through motor activity and by touching, seeing, tasting and hearing.  It may not seem that there is any mathematical construction going on during this time, however, children begin to make relationships between and among objects as they begin to construct ways to classify, seriate, compare, a order objects.  Classification takes a child's ability at matching objects and builds it into a system of organizing or classifying objects into groups with similar characteristics.  Classification is an important foundation for future mathematical concepts such as comparing sets of numbers and quantification.

           Beat and Music - Beat and music activities and materials are excellent for promoting emergent mathematics.  Using Bongo drums with infants and toddlers can help children experience the mathematics.  Teacher and child take turns repeating each others beat.  The teacher beats the drum twice, and the child beats the drum twice.  If the child takes the lead, the teacher can echo the child's beat.  This helps support a one-to-one correspondence relationship in the child.  It also supports a matching relationship which will refine the child's ability to classify.

          Use of synthesizers with an automatic beat generator is another good way to promote math through music or letting children play notes on the keyboard along with the generated beat.  These synthesizers come with headphones so children can play whatever they feel like and not bother other children in the classroom.

          I observed one teacher encourage her children to organize a marching band using the musical instruments and items in the room.  The children decided how to march.  One child even insisted that he say "one, two, one two" as they marched.  The children, for the most part, coordinated their beat as they marched through the hall of the center, outside and back to their room with one child saying "one, two" the whole time to keep them all together.

          Using numbers, counting, and quantification in everyday activities.  - Even children under the age of two can be exposed to math during everyday tasks and activities such as snack time or circle time.  Any opportunity to count should be taken advantage of to help the children make all types of relationships.  Teachers should count and use math whenever possible and even ask children questions about simple mathematical relationships.  This type of interaction helps children to recognize the importance of numbers and promotes the construction of emergent mathematics. Even children of this age can understand the concept of "more."  Asking children to compare groups of objects or quantities encourages the development of this relationship.

          Just because they have not constructed number is no reason not to use math around them.  Just as reading to infants and toddlers helps them develop literacy skills, using math around children helps them construct number concepts. 

          Blocks and Shapes - Children who are surrounded with interesting objects are naturally promoted to make relationships between those objects.  "Same and Different", matching, and classification relationships all require the child to focus on a certain quality of the object in order to make the comparison.  The more children make comparisons, the more complex their comparisons become.  The simple act of adding an increasing variety of colored balls or blocks to the child's choices can facilitate more and more complex mathematical relationships.  These activities support the concepts of seriation and classification.

          Construction using cardboard boxes can also help children make relationships.  In my experience, infants and toddlers love to play with cardboard boxes.  A variety of sizes of boxes can be made available for the children to stack and arrange to make structures.  Larger boxes can have doors or holes in them for the children to crawl in and out.  These boxes can be put together in a variety of ways and each combination or sequence is another relationship that the child has made.  In the process of arranging the boxes, the children would have some discussion and social interaction which will also promote the making of new relationships.

          Shapes can also be used in matching relationships.  In infant and toddler rooms there should be an abundance of different shaped blocks and tiles for children to match and compare.  Because their mathematical development is still in its early stages, infants and toddlers naturally look for exact matches.  This is the level of classification that they can handle.  They are unable to see something as "same" and "different" at the same time.  I observed a teacher working with a 12 month old.  They were examining a group of a blue and yellow triangle blocks.  The child gave a yellow triangle to the teacher and then picked up another yellow triangle and gave it to the teacher.  The teacher then picked up a blue triangle and showed it to the child.  The child grasped it and threw it back in the pile, found another yellow triangle and gave it to the teacher.  To the child, the yellow and blue triangles are not matches because they are different colors. 

          As children develop their matching and classifying skills, they will be able to make more complex relationships.  But these developments take time and interaction with objects and other people to construct.  Even if a human child is "prewired" for math, they still have to construct the concepts piece by piece.  Formal mathematics does not just appear; it is slowly constructed, step-by-step over the infant, toddler, and preschool years.  This is why it is so vitally important to offer children as young as a few months old opportunities to match, classify, and compare.

Three and 4 year olds

          As children begin to move out of their Sensory-Motor thinking and into what Piaget (1969) called the Pre-Operational stage, the big change is that children are able to think representationally and they begin to acquire a certain degree of abstract thinking.  Children can think about objects that are not right in front of them and they can begin to make relationships to previous experiences.  Children of this age can make much more complex relationships between objects.  This is important for emerging mathematical concepts because it is during this time that the mental structures that will allow a child to understand quantity are constructed. 

          The concepts of seriation, classification, and order take on a new dimension as children begin to be able to make more abstract relationships.  They can make comparisons to objects that are not present, or events that took place in the past.  This allows the child to synthesize order, seriation, and classification to construct abstract mental structures that will support quantification and formal mathematics.

          Children begin to make mental mathematical relationships that build on and refine the idea of  "more" into "one more" or "two more."  This refinement will eventually lead to the child being able to understand that "three" is one more than "two" and two more than "one."  This is the core idea behind quantification.

          Manipulative - An easy way to promote math in this age is simply to ask a child to use mathematical concepts in their activities.  If a child is using blocks, a teacher can ask "How many blocks do you have?" or "How many more do you need?"  Children are willing and even excited to count objects and make mathematical relationships if the teacher encourages them.  A four-year old child was making a chain out of different colored plastic links.  He was working alone when I asked him how long he was trying to make it.  He did not respond so I tried a more direct question.  "How many do you have so far?" I asked.  He continued to put on the next link and then proceeded to count each link.  There were eight.  After he put on another link I asked again "How many do you have now?"  He went back to the beginning and counted each link again and got an answer of nine.  When he again added another link, I asked him a more leading question.  "You had nine and you put one more on.  How many do you have now?"  Again he counted all the links until he got to the answer of 10.  After that I did not have to ask him again.  Each time he put on a new link, he would count all the links.  He eventually made a chain with 27 links.  However, after 15 his counting became erratic.  Sometimes he counted carefully and got the correct answer and other times he missed some links in his counting. 

          For example, after he correctly counted 26 and added one more, he counted again and missed a few.  After completing the counting he triumphantly announced "15!"  The fact that he now had less than just one time before did not seem to trouble him.  Even though he made mistakes and showed an incomplete understanding of number concepts, he is getting closer and closer to using mathematics in a conventional manner.  Just as children who move from drawing squiggles to writing words are learning to write conventionally.

          Every Day Activity - Just as with the infants and toddlers, everyday activities such as snack and circle time can be used to promote the usage of math.  Dividing up snack, counting plates, and other activities can be assigned to children.  They then have to use their own mathematical problem solving ability to figure out the best way to achieve the tasks.  A child who is assigned to put out the plates for his table of five may do it by going to the stack of plates, getting one plate and placing it in front of one child and then go back to the plates to get another plate for the next child and so on until everyone has a plate.  Eventually the child will realize that they can count the children then go to the plates, count out five plates, and distribute them accordingly.  Allowing the child to use their own methods of solving a problem such as this allows the child's emergent understanding of math to develop in a child centered developmental pattern.

          Assigning two children to figure out how to solve an everyday problem as described above promotes problem solving even more.  The children can discuss, plan, and even argue about the best way to solve the problem.  This argument will promote both children to construct new ways of seeing the problem (Kamii, 1990, 1991).  In an argument, the child must clearly communicate their ideas to another person and at the same time evaluate the other person's ideas.  In the process, the child examines and perhaps modifies their own ideas.

          The Project Approach -            The project approach to early childhood education allows children to explore their world and construct knowledge through genuine interaction with their environment. Lilian Katz (1989) states that young children should have activities that engage their minds fully in the quest for knowledge, understanding, and skill. When engaging in the project approach, the children are not just gathering knowledge from a worksheet, structured activity, or a teacher, but they are actively making decisions about not only what to learn, but how and where to learn it.  Through this method, children construct problem solving techniques, research methods, and questioning strategies.

          When children work on projects, a number of opportunities arise for children to use math.  In a recent project on construction and transportation, children had an opportunity to use measurement to help them build a truck.  They measured how long, tall, and wide they wanted it and then transferred their numbers to the cardboard they were using to make their truck.  Their measurements were not accurate, and they did not really understand the concept of using a measuring tape, but just as a child who writes squiggles on a piece of paper is learning to write, so these children were learning about measurement.  

          The children also learned about blue prints and when they made their own blueprints, the teacher asked them how many windows they wanted in their house, how many bath rooms, and how many rooms over all.  They discussed the lay out of the house, which rooms would have windows, and how the rooms were located in the house.  The children had to plan, count, use number and measure to complete the activity. 

          4) Voting - Whenever a decision needs to be made that the children can have an input on, voting allows the teacher to use math in an integrated way.   Not only does it offer an opportunity to count, but to compare numbers.  Children can be asked to vote on which book to read first.  The teacher asked the children to vote for each book.  As the teacher counted the hands, she encouraged the children to count with her.  If the vote was six to five, the teacher can ask the children which book had won.

Treating young children as mathematicians

          Lets review some of the basic ideas about young children and mathematics.

1.  Mathematics problems take time.  Allowing children to work for long periods of time on one problem encourages children to think as mathematicians. 

          As long as children are interested, let them work on problems until they figure it out.  They may fail a number of times before they finally solve the problem.  Children, like mathematicians, should be immersed in a complex problem that requires mathematical problem solving and complex numerical thinking.  This is one of the characteristics of how mathematicians work.  Good problems ask the student to figure out an innovative way to solve the problem without a time limit being set on their thinking process (Wakefield, 1997). 

2.  Children should be allowed to use their own methods for solving a given problem. 

          Andrew Wiles stated that doing math was like stumbling around in the dark for months until you find the light switch.  We should let children learn mathematics in a similar fashion. Children will understand the mathematical concept and procedure more thoroughly if they are allowed to use their own thinking process to explore mathematics as if it were a dark room and eventually find the light switch and come to an answer (Kamii, Lewis, & Jones, 1993).  It might not be the fastest or most efficient way to get an answer, and many children may come to the same answer in different ways but the children will understand the concept not just the rote procedure.

          In the process of discussing different procedures that other children used to get to the answer, the student compares the different methods to their own, thereby strengthening his or her understanding of the concept.  When students are encouraged to discuss different answers, they come to realize that answers do not come from the teacher but are universal.  In other words, 2+2 =4 not because the teacher says so but because the student has logically convinced himself or herself of the truth of the statement  (Kamii, Lewis, & Jones, 1991).

            For children to be treated as mathematicians, they must be expected to go through many wrong answers before they reach the correct one.  Emphasis needs to be put on the valuable mathematical thinking going on in the child's mind and not the answer.  It should be emphasized and modeled to children that there is nothing wrong with incorrect answers and that they are stepping stones to the right answer.

3. Children's Excitement comes from their own thinking ability. 

          Children can get very excited about a mathematics problem.   Children find pleasure and excitement in problem solving (University of Chicago, 1998).  If children are allowed to think for themselves, and discuss and defend their ideas, mathematics becomes just as fun as trying hard to complete a video game or working diligently to put a puzzle together.  This is illustrated in the following observation in a four year old preschool classroom.

Amy and Josh decide to share a package of M&M's.  They discuss different ways to split them.  Josh suggests making two piles that look similar but Amy suggests that each child take turns eating one candy until they are gone.  Josh is hesitant.  He feels that his method is better.  A five minute discussion ensues concluding with Josh conceding that Amy's solution is better.  They shake hands, giggle, and they begin to eat.

          The children did not proceed with the mathematics exercise because they expected to get a grade or a reward.  They used mathematics because it helped them solve a problem.  After they thought and talked about it they were happy with their decision.  The process was their reward. 

4.  Problems can have multiple solutions and many different ways to get to the solution. 

          Problems can and should spark discussion and even disagreement among the children trying to solve it.  An integral part of solving a problem is figuring out how to proceed toward the answer.  A good mathematics problem will have many different ways to proceed (University of Chicago, 1998). 

          The following problem is a typical type of word problem found in mathematics texts.  "Sara has 23 apples and Joe has 6.  How many apples do they have all together?"  A word problem such as this has a clear way to proceed and requires no problem solving on the students' part.  The student simply has to pull out the numbers, realize addition is needed, and use a standard algorithm.  However, the following problem is much different.

A recycling factory makes its own paper cups for canteen use. It can make one new cup from nine used ones. If it has 505 used cups how many can it possibly make in total?

          Here, the mathematics is quite easy, but the manner in which you proceed to the answer is much more complex.  The children may not realize the complexity of the problem the first time they attempt it.  The cups can repeatedly be recycled and, therefore, the problem becomes complex and requires problem solving ability to reach a solution.  The teacher could even give the students the answer, and they will still work on it until they understand how that answer could be right. 

          Children could discuss and argue this problem in groups.  They could make a case for a certain method, and answer and critique other student's attempts.  Children could also take this problem home, and ask a parent or sibling to help them.  This problem is just as difficult for adults to think about as it is for children.  Therefore, the child, sibling, and parents would all be on the same level of understanding.  For these reasons, these types of problem intrinsically motivate and excite children toward mathematics (Blake S., Hurley S., & Arenz B., 1995).

5.  Social interaction causes children to act as young mathematicians by requiring them to prove their answer and all the steps they took to attain the answer. 

          Social interaction is one of the most important parts of a mathematics program, especially one that views students as young mathematicians (Kamii, 1985).  However, it is the element that is most often absent.  Traditional mathematics lessons and homework are designed to be a solitary act.  Children are not encouraged to defend an answer or collaborate on solving a problem.  Instead they are given individual practice worksheets and asked to complete them quietly (Fosnot, 1989).

          However, if children are going to be viewed as young mathematicians, they must be allowed to collaborate, argue, consult, defend, ask, explain, and propose to, and with, other students using mathematical ideas (Kamii, 1985; Householder & Shrock, 1997).  Children construct mathematics understanding mentally, inside their head through this type of process.  Without this process, children just memorize how to get a certain answer without really understanding the concept.  Letting children use their own thinking helps children really understand math.  Rote teaching using the algorithm causes children to be able to perform and perhaps get the right answer but when children use their own thinking and explain how they attained their answers they will understand the concept underlying the answer more completely (Perry, VanderStoep, & Yu, 1993).

6.  The math is not in the manipulative it is in the child's mind.

          Many people assume that because Piaget's stages of cognitive development talks about "concrete operations", that young children cannot think abstractly and need concrete objects to do math.  As mentioned above, manipulatives are useful tools to help children think about mathematical relationships, but the math exists in the child's mind.

          Preschoolers can think abstractly to a certain extent.  According to Piaget, where preschoolers have some difficulty is in logical thinking.  Preschoolers, being in the preoperational stage, don't always see the need for every explanation to make logical sense.

Conclusion

          There are many easy things that teachers can do to promote the emerging mathematician in every child.  Questioning strategies, activities, and simple games offer a great opportunity for teachers to help children construct basic mathematical concepts.  An active stimulating environment and a teacher who is willing to see the child's ability to construct mathematical concepts is invaluable to a child's construction of mathematics.

          If we are to view the development of mathematics as emergent we must understand that construction of mathematical concepts begins the day that a child is born.  Children already construct the basic concepts of mathematics such as quantification, seriation, order, and classification without much interference or direct teaching from adults.  This understanding is not something that can be taught to a child.  They must construct it for themselves.  The role of the teacher is to facilitate this construction by offering infants, toddlers, and preschooler's opportunities and materials to promote their construction of mathematics.