Yes, all teachers confront this problem of addressing the ability levels of all their students. There is no such thing as a homogeneous group of children. There are always varying abilities, experiences and perspectives. Some helpful strategies that I and other teachers have used include: 1) Keep tasks open-ended. For instance, children might be challenged to create a design with pattern blocks that is symmetrical in one or more than one way. It's of course important for children to come together to share their solutions so everyone can benefit from each other's thinking. Another task might be: How many different ways can we classify these leaves we found on the playground? Still another: How many ways can we put these 5 acorns in these two boxes? These kind of tasks are accessible to all children, and everyone can feel successful. These problems all have multiple solutions, another feature of problems that meet the various abilities of all learners. 2) Pose questions that extend the given problem. In this way we are always challenging children to go beyond what they currently are doing. Such questioning demonstrates to children that problems can always be altered and modified in many different ways, and that one problem can be a seed for a myriad of related problems. Even a simple basic fact can be the catalyst for further investigations. Take for example 4 + 5 = 9. What do you notice about your problem? We might see that we added an odd and an even number to make 9. We could then extend the problem by asking, "Are there other ways to add one odd number and one even number to make 9? Or we might see the numbers 5 and 4 are consecutive. We might then extend that observation by asking, "Can we make 10 by adding 2 consecutive numbers? What about 11? 12? Are any numbers not possible?" So the questions come on the heels of some observations. What do we notice and how can we use that observation to pose further questions. This kind of strategy has been called the art of problem posing. 3) Individualize the task. I worked with a 2/3 grade class on multiplication. We read a book about sets of objects in the real world (What Comes in 2's, 3's and 4's by Aker) and the children then made a list of things that came in those sets, e.g. 3 sides on a triangle, 4 legs on a horse, 5 cents in a penny, 6 legs on an ant, and so on. We then asked the children to create multiplication books by using one of these sets as their basis, e.g. a child drew 1 ant with 6 legs and wrote, 1x6=6, then on the next page drew 2 ants, each with 6 legs each, and wrote 2 x 6 =12. We differentiated the task by asking the 2nd graders to pick a set that corresponded to one of the multiplication tables they were supposed to know by the end of 2nd grade (1-5) while 3rd graders selected one that was appropriate for them. In this way all the children were involved in the first part of the activity by responding to the story and adding ideas to our list of sets, and then worked individually on their own multiplication book.

My favorite math field trip is to the supermarket. SO much math there.

I do agree that we want children to use mathematics for authentic purposes. For this reason they need to be engaged in doing real tasks and activities that have a purpose and function. Cooking is an obvious example where children use measurement for an authentic purpose. Graphing is another important math topic that can be woven throughout the day and across subject areas. Children might create a pictograph about the weather for the month, or who has lost teeth, or the number of bird species seen at the class feeder, or which of several books the teacher should read aloud. It is also important that children have the opportunity to share their data with an appropriate audience. For instance, they might collect data related to school lunches and share their data with the cafeteria staff. They might collect data on which pieces of playground equipment are the most popular and present the data to the principal for consideration in future purchasing decisions. Having a real audience really stamps the experience with a mark of authentication.

Many classes go on a "shape hunt." Two excellent books that can launch this exploration are by Arlene Alda: Arlene Alda's 1, 2, 3 and Arlene Alda's A, B, C. Her stunningly clever photographs show the shape of numbers and letters in objects in the environment. For instance, a banana peel is curved into the shape of a 3; an "E" is found in the grooves between bricks on a sidewalk, and an "F" is shown by part of a railing on a set of steps. I worked with a teacher who read these books to her students, and the class then went around the inside and outside of the school looking for their own alphabet and number sequence. The teacher took photographs of what the children found and the children wrote about their shape discoveries and placed them in a class book. The children were encouraged to describe how the letters/numbers were the same and different, such as straight lines, curved lines, and open and closed shapes (and even lines of symmetry were noted by some children).

Children love to build. This kindergarten teacher had many photographs of buildings in the block corner to inspire the children in their building efforts. She also took photographs of the children's constructions and invited the children to write about what they did. This work was then placed in a class building book that was used by the children as a reference for future building ideas. In this way children were seen as legitimate builders in their own right.

Learning is about making connections. Planning integrated units of study that incorporate mathematics, science and social studies makes learning both meaningful and memorable. We know that learning is best understood and is more lasting when it is tied to authentic contexts. Fostering this interdisciplinary perspective in the early years is essential for young children's learning. Children view and experience the world wholistically and so integrated learning makes perfect sense to them. However, we have learned that the more we fragment learning into separate cubicles of math, science, etc the more we make learning hard.

The building corner is a great place to begin thinking about integrating math and science. You might start by looking at what kids are already doing there. How are they incorporating mathematical and scientific skills and concepts into their work in this area? I have seen the block corner turned into a store front as children learned about stores in their neighborhood, and how local stores buy and sell goods. Cooking on a regular basis is a clear integration of science and mathematics as well.

It is useful to have at least one ongoing investigation throughout the year. In this way children can really have the time to explore in a very in-depth way. I was involved with a teacher who set up a bird observation corner that became a popular spot for children to use all year. There were stick-on feeders on the windows so that children could see birds right up close. The children kept a class journal of their observations and graphed the species they had been seeing. They even participated in Cornell University's Project FeederWatch. The children sent in their data about what they had been observing and became part of this national effort by amateurs to document changes in bird populations. In another classroom the teacher invited the children to plant and harvest mung beans to feed their classroom rabbit. There was obviously a lot of weighing and measuring involved in this experience, as well as learning about the growth of plants.

I think that the most important thing that early childhood educators need to do when it comes to STEM is to promote the natural curiosity in children and to stimulate a child's to explore and ask questions about everything. To do this we need to step away from the canned math and science curricula that seems to be so prevalent in schools. Unfortunately, more times than not, school actually discourages students from reveling in the wonder of math and science. Children begin to see math as just a lot of rote memorization and science as a bunch of terms that need to be memorized.

In other words, sometimes the best thing we can do is to get out of the child's way. The worst thing that a teacher can do is to over-control. Instead of planning teacher directed "units" and "lessons" with closed ended and definitive outcomes (i.e. children spend the whole lesson following a set of directions and reaching the predetermined outcome as planned by the teacher/textbook), plan child-centered projects that allow children to investigate and come up with their own questions and find their own answers. Facilitate exploration rather than teaching them an isolated "fact". Teachers who can follow children's interests and build exciting long term investigations using the project approach will find that the questions (and answers) that children come up with are far beyond anything that a textbook or canned curriculum could ever hope to achieve. It does mean that there is some uncertainty for the teacher on a day to day basis. With a preplanned 2 week unit, you know exactly what lesson will be taught and what the child is expected to "learn" on a specific day. With a project you cannot always predict where the children's curiosity will take you, let alone how long it will take (2 weeks? 4? 6?). Each day, the teacher needs to evaluate what happened that day, review the childrens questions and interests and decide what to do the next day.

Children (especially young children) have some interesting hypotheses about how the world works. Instead of telling them that they are wrong and then telling the "correct" answer, we should be encouraging and facilitating them to think about their idea, test it and reevaluate it. If I could use my 4 year old as an example again, this morning he yawned and then coughed (we have both been battling colds for over a week). Because of this he pronounced that yawning caused you to cough. I asked he how he figured that out and he said "Well I just yawned and then coughed, so yawning causes the cough". Now, I could point out that germs cause coughs or that it is allergies but instead I chose to wait. The next time he coughed I asked him if he yawned before he coughed that time. "NO! I didn't! Maybe its not EVERY time."

As he uses his natural curiosity to think about this I am sure I will hear more of what he finds and how he chooses to explain the relationship between yawning and coughing.

So in answer to your question, the best thing we can do is stay out of their way and facilitate their wonderful ability to think by questioning.

I know teachers who have a weekly newsletter that is sent home to parents. This newsletter can be a regular way to communicate to parents about mathematical projects and activities in the classroom as well as an invitation for parents to share about the mathematical learning they see in their child at home. As the school year progresses the newsletter becomes owned by the parents more and more as they share their observations with others. The teacher can also describe some useful activities to do at home and give reasons for why these activities are important. It is this explanation of the activities that gives parents an insight into the development of their children's thinking. For instance, a teacher might suggest children be involved in sorting clothes as they come out of the dryer. She can then explain the importance of classification experiences. Classification is a crucial strategy for young children to understand before meaningful number work can begin. Before children can group objects they must know what a group is. Sorting and classifying can help children build this understanding. By sharing this developmental perspective teachers can help parents appreciate even more their child's intellectual development.

Every program has different parent dynamics and the first thing that any teacher needs to do is to understand what their parents are like. What is their educational level? What is their comfort level with mathematics? Do the children have a lot of siblings or just a few? Do they tend to be older siblings or younger? Interestingly enough, the fathers educational attainment seems to have an even larger impact on children's later mathematical achievement, so knowing how present the father is in the home would be beneficial to know. It also tells us that if we want to improve long term mathematical success we need to not only involve parents and families but reach out to fathers.

Having materials that can be taken home is a good place to start. I wrote earlier about math games in the classroom, however these could be modified into "take-home" versions. Collect old lunchboxes and put all the materials and a sheet of instructions in each box. Children can be encouraged to check out the boxes for a week or so just like they check out library books. In your communications with the child's parents, let them know how important it is for them to play these games with their child in the evenings and over the weekends. This type of family involvement helps parents to interact mathematically and also to recognize the progress that the child makes as they play the game over and over again.

You can also organize family math nights. These are like your typical open house, but there will be many games and activities set up around the room for parents and their children to interact with. You could also plan for a group activity or game that involves math. The goal is to have a group activity that involves the parents and their children in activities that involve math is a fun way.

Time is always a consideration and it is one of the key things I hear from teachers whether they have 2 hours or 8 hours. Finding time in the day for math can be hard, but if you make math a priority in your classroom, it becomes a little easier to plan around it as part of your day. Most programs already make literacy activities a priority and set aside specific time to read books and do other literacy activities. However this also does not mean that you have to set aside 30 minutes each day for math.

Especially in preschool (and infant and toddler) classrooms, math is learned best when it is integrated into other activities. When reading books to children it is quite easy to find things to count or compare or to look for patterns. Same thing with music and singing, physical activity and imaginative play. My 4 year old even found math while watching TV last night (yes, I do let my child watch TV occasionally). He began counting the commercials and he noticed that there were 3 commercials. After the next commercial break he said "There were 3 AGAIN!". To which I said "How many do you think there will be next time?". He said "I think there will be 3. Lets see."

All that is really needed is for the teacher to listen and pay attention to the childs natural curiosity and to tweak that curiosity occasionally. It is not necessary at this early age to have long math lessons or group activities focused specifically on mathematics. You can use the activities you already have planned and look for the math from within that. Singing, Music, Storytelling, dancing, physical play, imaginative play, blocks etc. . .

The math is already there, teachers just need to become more aware of it.

Here are some suggestions of what teachers can do from a previous article I did for YC:

• Provide plenty of blocks and other toys and items of different shapes, colors, and sizes.
• Play with children, notice what they do, and record observations, support their play by asking questions and promoting interactions with objects.
• Use words that describe attributes such as size, shape, and color: “You made a big pile of blue blocks.”
• Provide plenty of sound makers (e.g., wrist bells, pots and wooden spoons, rhythm instruments) so children can experiment and experience rhythm and steady beat.
• Encourage children to play and move along with recorded music.
• Talk with children and describe what they are doing: “Shake, shake—shake, shake, shake. You made your own music.”
• Use a steady beat when rocking infants perhaps add a chant to it
• Rhythms can be incorporated into the rocking or bouncing the child (i.e hard bounce, soft bounce, hard bounce, soft bounce)
• Set up a simple dramatic play area with many props that encourage children make one item “stand in” for another.
• Provide a variety of toys that invite children to explore with their senses and motor skills and allow them to compare and contrast objects by size, color, texture, and sound. Some good toys for this purpose include xylophones, stacking rings, shape boxes, and texture balls or books.
• Point out mathematical and relational comparisons during daily activities. For example, serve two kinds of fruit and say, “These apples are hard and crunchy. The bananas are soft and mushy.”
• Introduce mathematical words to children in matter-of-fact ways: “These blocks are longer than those blocks.” “These are square and those are round.”
• Encourage children to explore how their own bodies fit in space and to see things from different perspectives (e.g., inside and outside, high and low). Provide an expanding tunnel or one made by taping together several cardboard boxes.
• Let children climb on a stack of soft pillows. Talk about what children are doing so they can begin to learn the words that describe mathematical concepts: “You were in the box, then you climbed out.” “You climbed up on the pillows, then you jumped down.”
• Offer materials such as sand and water (or other safe materials) and containers of different sizes, shapes, and capacities. Allow children to interact by filling and emptying the containers and noticing what happens. Teachers can focus a child’s thoughts by asking questions such as, “What might happen if you pour that into this jug?” or “Do you think all of the sand will fit in this bucket?”
• For infants simply giving the infant a ball for each hand to hold will help them to construct concepts such as “more” and “one”

This is a question near and dear to my heart and I am working on a book on this exact topic, so you will have to excuse the length of this answer. Ready? Here we go!

Most people, even those who claim to hate math, actually enjoy solving problems and puzzles. Crossword puzzles and Suduku puzzles are a staple in most newspapers. I can hear your thoughts now, you don’t believe me.

So I understand that if at this point you are thinking to yourself that math is not fun and never will be. However there is ample research to show that humans are born with a mathematical brain. Our ability to use mathematics to order and structure our world is pre-wired directly into our brains. Again, at this point you are thinking to yourself “Not MY brain!”
Math anxiety is not born from an inability to do math, but rather from the way we are taught mathematics in 12+ years of school. It is not the mathematics that causes the anxiety. It is the fear of failure and the misconception that mathematics is about correct answers. Answers to math problems are the destination, but math itself is more about the journey. Sir Andrew Wiles is a mathematician famous for his solution to a 300 year old problem that for all that time, had no answer. But that did not stop him from devoting is career as a mathematician to is solution. In an interview he said:

I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days. I loved doing problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem—Fermat's Last Theorem. This problem had been unsolved by mathematicians for 300 years. It looked so simple, and yet all the great mathematicians in history couldn't solve it. Here was a problem, that I, a 10 year old, could understand, and I knew from that moment that I would never let it go. I had to solve it. . . Pure mathematicians just love to try unsolved problems—they love a challenge.

Many early childhood teachers feel uncomfortable teaching mathematics because they did not and do not like mathematics. Many also feel that they are not good at mathematics and therefore feel uncomfortable teaching it to their students. Math anxiety is a well researched topic (Stuart, 2000; Levine, 1995; Burns, 1998; Altermatt & Kim, 2004) and current practices perpetuate the problem. Many teachers who have math anxiety themselves inadvertently pass it on to their students. If you have math anxiety, rest assured that you are not alone.

Math anxiety does not come from the mathematics itself but rather from the way math is presented in school and may have been presented to you as a child (Stuart, 2000; Wiebe et al., 1987; Third International Mathematics and Science Study, National Center for Education Statistics, United States, & Office of Educational Research and Improvement, 1997; Seeger, Voigt, & Waschescio, 1998). Often math is presented as a high stakes memorization process. Think back to your early school years, such as second grade. Do you remember timed tests? Many students are subjected to these high stakes competitions meant to help children learn their “math facts.”

I can personally remember a chart posted prominently in the classroom with all the students names in a column down the right hand side of the chart. As we progressed through the year, we had daily timed mathematics tests on addition (or was it multiplication? I can’t remember). If we completed all 20 problems in 1 minute we got a star next to our name and got to move on the next level test. If you did not finish in time (with all the answers correct, of course), we got no star and had to retake the test the next day and subsequent days, until we passed it and finally earned our star. Near the middle of the year, everyone could see, by looking at the chart, which students had more stars and which students had the fewest stars. As you can imagine, those of us with the fewest stars began to really hate math and really stress out whenever it came time for the test.
So math anxiety is a common problem and it is caused by the way we were taught mathematics - what can we do about it? The first thing is to recognize the signs of mathematics anxiety in ourselves. Then take a deep breath and remind yourself that no matter how scared of math you are, you still have enough mathematical content knowledge to teach children up through age eight or nine. The second thing to do is to make a pact to yourself, that as a teacher you will do your best to fight math anxiety in the children that you teach. Fighting math anxiety is not hard to do because it is, for the most part, created by the way we teach mathematics in the first place. By making mathematics about process rather than product, integrating mathematics into everyday activities, having children use their natural mathematical ability, making mathematics relevant to children’s lives, and , above all, avoiding high stakes and high stress activities and assessments (such as timed tests), we can eliminate the underlying causes of math anxiety and may never have to hear “I hate math” ever again. Children are naturally drawn to mathematical and logical thinking so all we have to do is get out of the way and make sure we are not ruining it for them.

Using Reflective Practice:

Teachers need to understand this behavior. One way to do this type of self-examination is through the reflective process. Reflection is one of the most important aspects of teaching. You must be able to look at your own decisions and abilities in relation to your math teaching tasks. Tooke & Lindstrom (1998), Harper & Daane (1998), Godbey (1997) all found that pre-service teachers come to their methods classes with high levels of math anxiety. Methods classes seemed to mitigate these anxieties at least short term, but our goal as teachers should be to find ways not to pass on our math anxiety to our students in the first place. To begin the process we have to examine our practices through a reflective process. The words over the oracle at Delphi read “know thyself”. Good advice for us as teachers too.

Many may have engaged in reflective journaling such as a diary or daily notes, but true reflection much more than just writing down what happens in the classroom and what you do and see and feel. Reflection is a kind of problem solving that once the basic journaling ends. Why a lesson presented in a certain manner? What theoretical base was the lesson based on? How would a teacher with a different philosophy teach the same lesson? How does the lesson effect the anxiety level of the student?

The next step in reflection is to act on your observations and interpretations, for example, “How can I teach that same lesson without raising the anxiety level of the student?” or “How can I teach an interactive lesson and still prepare the children for the standardized tests?” This is the essence of reflective practice.

Teaching is all about making decisions (Fosnot, 1989). Teachers need to be able to assess a situation and make complex decisions based on a vast knowledge base developed during their teacher preparation program. Teachers as decision makers must become adept at combining what the children need to know, what their knowledge base tells them is the best way to present this knowledge, and the needs of the individual children to create effective and enjoyable learning experiences. Information about child development, their class, their individual students, and themselves all go into making a decision about how and what to teach. Teachers must develop a knowledge base from which to answer these questions and make the best decision possible. A knowledge base includes all the things that you learn in your teacher preparation program courses such as Child Development, Guidance and Discipline, Teaching Methods, and Developmental theory just to name a few. As you learn this material in your classes you are developing your knowledge base, and when you become a teacher you should be sure to use this mass of knowledge to guide your curricular planning process in all your subjects.

For example if you are planning a mathematics lesson, the first thing you need to think about is your objective for the lesson. Is it to teach the children to learn a specific skill, or is it for them to understand the concept behind that skill? This decision should be informed by your understanding of developmental theory and child development and will have a great impact on how you design your lessons. One part of that knowledge base is a theoretical foundation. To be effective, teachers need understand different theories of child development and current research on how children develop. An understanding of theories past and present can help us evaluate current practices and improve those practices for the future.

As for the “what” in “what children need to know”, that is a bit more complicated. Deciding what children need to know and when to teach it is what doing a decision maker is all about. Each individual state has a different set of standards that must be met and local school districts often have their own requirements on top of that. In many states, textbooks are adopted at a statewide level and in others it is done at a local level. So deciding “what” to teach needs to be based on what you know about children and the way they learn mathematics. Actually, using the word “decide” is a bit misleading also. Most of the time you should let the children show you what they are ready to learn. So in a way they decide for you (if you pay attention).

So as you are preparing to become a teacher you should ask yourself, “What kind of teacher I am going to be and how can I be sure to to add additional stress and anxiety to children’s lives.?” When you do field work or observations in schools you should be asking yourself, “What decisions has the teacher made to develop this math lesson? What were those decisions based on? What theories can I see behind that lesson? Whose research does that activity represent? What developmental principles is that based on?”

Teachers make decisions everyday, decisions that go beyond simple training, what has been done in the past, or what the teacher’s manual says to do. If teachers are to manage the level of anxiety in the classroom, they need to use reflection, know the research and make good decisions about the curriculum. Many times the level of children’s anxiety is not taken into account when developing mathematics curriculum for young children, or any other subject area for that matter. By recognizing our own limitations and thinking about how our decisions affect the emotional state of the child we can create a classroom where learning mathematics is not a source of anxiety.

Try to help your teachers to see that mathematics is everywhere in their classroom. Instead of directly teaching children their numbers or how to count to ten, point our how many fish are in the fist tank and encourage them to count them whenever they can. Same thing is true at snack time. Have the children take turns giving everyone a napkin for lunch. Ask them questions like "how many do you think you will need?" If the child says, "I don't know" then you might ask them how they might be able to find out. This type of questioning stimulates the child to think and problem solve mathematically without formal "lessons" on mathematics.

Games are another great way to introduce mathematical concepts. Many games can be teacher made with recycled materials. Dice, and spinners can provide some mathematical element to the game. An easy one I used to use frequently to promote counting and one to one correspondence is to take an egg carton, come counters of some sort (any small object will do) and either one or two dice. The child rolls the die or dice and then puts as many counters in the egg carton as they rolled. So if they roll a 4 they fill f holes with one counter each. Children can play alone or against another child. First one to fill up their tray wins. Keep rules to a minimum. What is also interesting is to watch the extra rules that the children make up on their own when allowed.

For older children, I like card games like the old stand by "War" (if you object to the violent name, feel free to change it but this is what I grew up calling it). Children divide the deck between them (you may want to take out the face cards beforehand). Even though dividing the deck is not part of the game, it does involve math so, again, try not to over control. Let them come up with methods of splitting the deck. The teacher can ask "Are the two decks the same?" or "How do you know you both have the same number of cards?", but it they are satisfied, don't interfere. If it becomes an important point later, you can use it as a "teachable moment". Once the cards are divided, each player turns over a card at the same time. The one with the highest number wins and takes both cards. If the cards are equal, they have a "war". The rules can be different from class to class and may even evolve as the children learn the game. You can also play "double war" where each child puts down 2 cards and the one with the highest sum wins.

Path games that you make your self or modify from store bought versions are great too. Candyland using dice or a spinner was a very popular one in my classroom as was Chutes and Ladders.

Music is a great way to promote math in the classroom, but I will let you read about that in the book, since I have taken a lot of space here.

Thanks for a great question!

Touch counting is a rote way to add. It does not teach about number relationships. However, this is not to say that children can't learn some interesting mathematical relationships with their fingers. For instance, one 6 year old used her fingers to show me the following: "Look, 2 and 2 is like 3 and 1." She held up 2 fingers on one hand and 2 on the other, and then lowered one finger on one hand and raised one finger on the other hand. This is the law of compensation in action! If I decrease one addend by a certain amount and increase the other addend by that same amount, the sum remains the same. As children get older they can use this same strategy on a problem like this: 98 + 47. They can add 2 to 98 and subtract 2 from 47 to find: 100+ 45 is 145. Teachers can also extend this child's observation by asking, "I wonder if you could show 4 in another way. How many ways can you do it?" In this way we are encouraging children to create equivalent names for a number. This concept of equivalence is one of the essential concepts in elementary mathematics.

Another 6-year old held up 3 fingers on one hand and 1 finger on the other hand, and then crossed her arms to show the numbers in reverse order. She proudly proclaimed, "Look, 3 and 1 is the same as 1 and 3!" What a lovely demonstration of the commutativity principle for addition, e.g. that you can reverse the order of the two addends and the sum remains the same.

Another 6-year old child held up 1 finger on one hand and 1 finger on another hand and proclaimed, "See, 1 and 1 is 2!" The child then proceeded to hold up 1 more finger on each hand, and counted aloud, "And 2 and 2 is 4, and 3 and 3 is 6, and 4 and 4 is 8, and 5 and 5 is 10!" What a beautiful demonstration of even numbers! And by showing it with two hands she nicely shows the symmetry of even numbers. This finger demonstration can lead us to explore even and odd numbers with the children.

The use of fingers can also nicely show how the numbers 1-10 are related to the anchors of 5 and 10. For instance, as a child holds up 5 fingers on one hand and 1 finger on the other, he is showing how that sum of 6 is related to the anchor of 5 plus 1 more. And 7 is then seen as 5 + 2. Or we could relate the 7 to 10 (another anchor to relate to), and say 7 is 10 minus 3 fingers. Then 8 is seen as 5 +3, or 10-2, and 9 is seen as 5 + 4, or 10-1. In this way the counting numbers are related to one another and not seen as separate entities. In this way numbers are related and connected to each other.

So my point is that fingers can be a useful way to explore number patterns. Look to see how your children are using their fingers and build on what you see are mathematically rich opportunities. One last note: counting on by using their fingers is not a strategy to be fostered. If you see children using their fingers to solve 7 + 6 by starting at 7 and then counting 6 more you know that they have not devised any effective strategies. They can use counters to see how they can relate this fact to doubles, "Oh, it's like 6 + 6 and then 1 more;" or "It's like 7 + 7, and then 1 less;" or they can see 5's inside each number and say, "I see 7 as 5 +2, and 6 as 5 +1. So I can add the 5's to make 10, and then add 2 + 1 and get 3, and then add 10 + 3 to arrive at 13."

We want children to learn basic facts but we want them to know with understanding. Using fingers and counters can help to underscore how facts are related.

I promise to answer your question, but first let me rant a bit.

I don’t think children should be “taught” any specific way of doing mathematics, especially in the early grades. Constance Kamii’s work shows us that math is reconstructed in each child’s mind by being actively engaged on a mental level. One of the problems that we all have with math is that we think about it as a collection of “facts” and “skills”.

Teaching approaches that rely on “Facts” and “Skills” focus on teaching a child to get “correct answers” and suggest that math is about memorizing “facts” and practicing “skills” over and over. In fact constructing mathematics concepts is much more like play. Give a child a new toy and watch how many different ways they find to use it, experiment with it and share it with other children. Think of math in the same way. You can give a child or group of children a math problem as simple as 4 +4 and watch them play with it, share it, discuss it and ultimately solve it in many different ways. The teacher/adult role is to question and facilitate not to dictate or "teach" in a traditional way. The more an adult intervenes to tell the child how to do the problem a certain way, the less a child learns to rely on their own natural thinking ability.

For this reason, I avoid using the words “skills” or “facts” or even “taught” when describing the process of learning mathematics, especially in the early years.

OK, now to answer your question: We should not teach children to use any one specific method of doing mathematics, but rather we should encourage children to use their own natural born ability to think to solve problems in ways that make sense to them. If they want to use their fingers, great. Blocks? Wonderful! Pencil and paper? Stupendous! Toy trucks and cars? Why not!

We should avoid teaching children “tricks” that help them to simply get answers and that short-cut the thinking process. Touch math, to me is one of those “tricks”. Also, surprisingly enough, is the standard addition/subtraction algorithm which teachers a handy-dandy trick for carrying and borrowing, but also never lets the child think about place value or construct other interesting and inventive ways to get to the same answer. When we teach the algorithm “trick” children never use their natural ability to think to solve the problems (and when they do they are scolded by the teacher for not doing it like they were taught).

Some big math ideas that are important to weave throughout the early childhood years include: patterns, data collection and analysis, number relationships, equality/inequality, and place value. Discovering patterns and relationships is especially important because it capitalizes on children's sense-making abilities. Those abilities are the most precious gifts that children bring to us in school and we must use those so that mathematics becomes not a mysterious system of rules to follow but rather an interesting avenue for exploration.

Learning math is an emergent process that begins at birth and should be supported by offering children a stimulating mathematical environment that encourages children to use their own natural curiosity and ability to think.

It is never too early to promote mathematics with young children as long as it is don’t is a developmentally appropriate way. In fact at age 4, your child has already begun to construct a lot of mathematics. The important thing to remember is that we are not talking about math using pencils, paper, worksheets or even traditional “problems”. Instead we should be creating stimulating mathematical environments for young children. Talk about math whenever you can. Ask children to count things at the grocery store or when driving in the car. Ask them to compare properties of objects such as size, shape, weight, color and even quantity. Ask “how many” about anything you can. Point out patterns and encourage your child to do the same and to create patterns out of everyday materials.

Mathematical development is much like literacy development in that we can support it very early in the child’s life. We know that the more children are read to, the better their literacy development later in life. When you read to an infant, they may not recognize the words on the page or even understand the spoken words, yet reading to them is a beneficial activity that leads to development of both of those abilities. Math is the same. Whenever you have the opportunity to ask about mathematical properties or have a child count and even add, take it. If the child responds with a wrong answer, don’t worry too much. I would suggest not even correcting the child. Instead ask the child “Why do you think that” or “How do you know that”. The richness of the child’s answers will amaze you. I sometimes even challenge a child with a different answer just to have them explain and defend their own answer.

Here is a dialog I had with my little boy this weekend. He just turned 4 in June. On Thursday he came to my class and told one of my students that 4 plus 4 was eight. My student was amazed so as my wife and I were getting ready to put him to bed Friday night, I said “Tell Mommy what 4 plus 4 is”. This is the dialog that followed.

Mike: “4 plus 4 is eight”
Me: “Is it? You know, I think it is 9”
Mike “NO DADDY!! Its 8. See [holds up 4 fingers on each hand and counts as he lowers the fingers first on one hand and then the other] 1, 2, 3, 4, 5, 6, 7, 8! See it’s 8!”
Me: “Oh. OK, I see”
Mike: “And 5 plus 5 is 10. See [Does the same thing as above but he adds the thumbs to make 10] 1, 2, 3, 4, 5, 6, 7, 8, 9, 10!”
Me: “Wow. You sure know a lot of math”
Mike “Yeah. And 4 plus 5 is 9 [He counts again] 1, 2, 3, 4, 5, 6, 7, 8, 9! So you were right about that one daddy [referring to my earlier attempt to get him ti think 4 plus 4 was 9].
Me: “I think you are right”
Mike: “Yeah and 5 plus 4 is nine also”
Me: “So 4 plus 5 is 9 and 5 plus 4 is 9?”
Mike: “Yeah” [At this point our 12 year old, who was listening from the kitchen yelled up the stairs]
Dylan: “That’s called the associative property!

We have never drilled Mike on any “math facts” or used flashcards or any math program. We have only encouraged him to to count, make relationships, and see the math in the world around him.

So you can involve your younger child in the mathematics by involving the whole family in these activities. Use mathematical language and point out the math you use when cooking, setting the table for dinner or when taking a walk.

Making the math relevant is essential for these learners as it is for all learners. What are the topics that they are always talking about and how might teachers use those as springboards for some mathematical investigations. For instance, I have known teachers who have invited older kids to figure out the cost of owning a car, or taking an imaginary trip somewhere, or figuring out the "best deal" when buying a cell phone, and so on. Other good math topics can be found in the newspaper. I have known teachers who have read an article in the newspaper on a regular basis using various sections of the paper (sports, arts, business, politics). They then use that article as a basis for exploring some of the mathematics.

I have also found the use of manipulatives to be effective. I tell older learners, "These blocks might seem like materials for younger students, but they aren't. We can use them ourselves to challenge our own thinking." As long as I acknowledge this feeling at the beginning so the students don't think I am treating them in disrespectful manner, they are willing to give things a try. I have used base 10 blocks to explore place value, and square tiles to build rectangles to show multiplication facts.

I have also found it helpful

It is hard fixing problems that were caused so early in a child's schooling. Our traditional method of teaching formal mathematics is through a skill and drill approach that emphasizes "skills" and "math facts". Children learn to memorize set problem solving steps rather than by truly learning to think mathematically about a problem. What happens is that as children get older they develop a dislike for mathematics and a problem seeing the relevance of mathematics. I know because I was one of these children. I am sure my High School math teacher felt about me the same way you feel about your students.

My problem was that I gave up on mathematics in 2nd grade when I struggled with timed-tests. I began to fear those tests because I knew I would fail at them. Math anxiety and a view of mathematics as a bunch of things I had to memorize killed my love for mathematics. I am not alone in having a story like that. But I also see those same people that love to solve Sudoku puzzles or who will secretly work on a "brain teaser" question - as long as there is no one judging them (except themselves).

So my answer to you is to try to make math relevant to them. Use games and puzzles to engage them in mathematical concepts rather than just teaching them short-cut algorithms to quickly get to the right answer.

One more trick that I will suggest will feel very unnatural the first time(s) you try it and will take a lot of self-control. When you pose a problem, whether it is to your class as a whole or to a small group of students, do not give them the answer or tell them that their responses are right or wrong. I know this sounds very radical, but it will change the very nature of how your students think about mathematics.

Pose a problem then let students think about it and suggest possible answers. As they do so, do not make any comment on its merits. Simply record the answer (on the chalk board perhaps) and then ask if anyone else has a different answer. When you have collected a few different answers ask your students to explain their answers. You may facilitate discussion by asking questions that help the student to explain their process better, but stay away from judgments. The more that you refrain from telling them where the errors are, the more the students talk to each other and find those errors for themselves. They will begin defending their answers and coming up with different ways to explain their answer and why they think their answer is correct. Eventually answers are narrowed down as flaws in process are discovered. Eventually you may get to just one answer that they all agree on. And I have never been in a situation when that one answer left standing was not the correct one. Your students will still look to you to validate their answer. You should refrain from doing so. As soon as you tell them the answer that you think is right, all thought on that problem stops. If you do not tell them an answer, the curiosity may continue. They may take the discussion home, or talk with friends about it.

I find the videos made by Constance Kamii in the 80's and 90's about children using constructivist methods in the classroom to be a great example of this process. You are in a situation where you are rebuilding a persons attitude toward mathematics, not just their content knowledge. Your mission is to help these students to focus on the process of mathematics rather than the memorized content that that have struggled with all though their schooling.

Toddlers and preschoolers understand more than you may think. Yes, counting is not always linked to mathematical understanding of quantity, but it is a step in the right direction. Jean Piaget used a number of logic problems to examine how children think about issues such as mathematical concepts.

The best thing that you can do is to use questioning techniques with children and introduce a number of different mathematical activities and games into their environment that promote interaction with objects and quantities. Teaching children to count may not directly teach the concept of counting, but giving children lots of objects to sort into groups and compare will help them to construct these concepts. Interactive mathematical activities where children are mentally stimulated to think numerically will lead to concept construction. Direct rote teaching children to "say their numbers" or "count to 10" will not teach these concepts.

I like to use the setting the table as an example. For each person their needs to be one chair at the table. There also needs to be one napkin, one spoon, one fork, one plate etc...

In essence, one-to-one correspondence is about making equal sets of objects in a very primitive way. For adults and older children we count the number of items in the original set and recreate it by counting out that many for our "new" set.

A professor of mine used the place setting example to demonstrate this. A 4 year old was asked to be in charge of making sure that everyone had a napkin. There were 4 people in the family so 4 places at the table. The first time the child looked at the table and went to the cupboard and got one napkin and put it at one place. She then went and got one more and put it at the next place. She did this 4 times. She did this every night for a long while. Then one night she decided she could count. So she counted "1,2,3,4" and then went to the cupboard and counted "1,2,3,4 napkins" and then placed them around the table. This she did from that night onward until there was a guest one night. The little girl counted out her 4 napkins and put one at each place. But there was one place without a napkin. As adults we would just go get one more napkin. But this child took all 4 napkins back to the cupboard and went back to one-to-one correspondence. She got one napkin and put it on the table and went back to the cupboard 4 more times until each had a napkin. The next night when there were only 4 again, she still made 4 trips to the cupboard and did every night after for some time until she again felt comfortable counting the napkins and taking them to the table. The next time there was a guest, she was not confused. She just counted 5 instead of 4.

So one-to-one correspondence is a transitional point and an important milestone to more advanced ways about thinking numerically.

The kitchen is the best place for me. There is just so much math there. I have my 4 year old help me to decide what we need to buy at the store and think about how much we might need. If we are thinking about meal planning I might ask him how many days we need to plan for. He may say 7. I could then say "Well what if we go out to eat on Tuesday? Then how many days?".

Our trip to the market is also filled with math. From how much things cost to how many apples we need to buy (especially if we need one apple for each day). We also weigh things and count the eggs in the carton. When we get home, cooking is filled with mathematics.

Math at home is quite easy all you need to do is look for the opportunity to have your child count or use quantity to make decisions. I also think I may have given some other good examples in previous answers so you might look there too.