My name is Betsy Silverman and my background is in hospitality and in design. During this phase of my life, I am mommy to Max, a 5 year old boy who loves to pretend. I'm trying to parent in a way that encourages imagination, environmental consciousness, and creativity. My plan is to look around and try to see with Max's fresh eyes and open heart. My goal is to try to see possibility instead of preconceptions and labels. Then, I hope to take things that we would normally cast aside, and imagine it's possibilities. I hope you will find some of my ideas interesting and hope that they will inspire some ideas of your own. Thanks for reading!
I’m good at some things, but math is not one of them. My trouble with math is not for lack of trying – I got extra help in school and even had a few outstanding math teachers. My brain just doesn’t work that way. So when it came time to teaching math to my 5-year old son, Max, I had no idea of where to begin. I began to introduce counting and numerals. My plan was that when he mastered counting 1 through 10, I would then teach him to recognize the numerals. When he did that too, he would then be good at math. I now know that plan was ridiculous. Numerals are not math; numerals are an abstract representation of quantity.
Wikipedia defines math as the study of quantity, structure, space and change. It says that mathematicians seek out patterns to form and prove new hypotheses. So math, especially preschool math, is not about recognizing numbers or being able to count. For me, this was shocking.
Did I waste my time trying to teach Max his numbers? No. Numbers are important. But consistently identifying numbers doesn’t start to happen until around age 4. It happens as children need to know it to be able to communicate about and understand the things they want to do as part of their day. So, instead of frustrating Max by drilling him on numeral recognition, my initial math energies should have focused on things that his young brain is designed to learn.
Max is just 5 so I hope I haven’t blown it completely, brain-development-wise. But if I had to do it again, my energies would have been better spent working on sorting, sequencing, patterning, number sense and estimation. This article is (part one in a series of 3) about the stuff that I should have been teaching and I would have done it without buying lots of math “games” and flash cards and puzzles. So, armed with the toys I have anyway, and the trash my family creates, my plan is to foster Max’s mathmatical learning before he enters kindergarten.
The purpose of this sorting game is to help children focus on one sorting characteristic. Obviously, as your child gets older and better at the game, you can add a level of difficulty by having them sort 2 or 3 characteristics.
To make it easier for your child (nanny or husband) make picture labels for your bins.
Max likes to help sort the recycling
And, if nothing else, your errands will get done and your child will be entertained.
Max likes to sort screw bolts and nuts for his “construction projects”
If you have a shoeless house, like we do, you probably have a huge jumble of shoes by the door. Ask you child to help you by matching up the pairs of shoes. I was amazed that Max not only did this but then arranged them by height!
Anything can be arranged in order.
For children who might not be completely sure of their numerals, cut a box up into squares or circles that get progressively larger and put numbers on them that get larger as the paper size increases. The size of the paper and numbers allow them to use either their knowledge of relationship of size and helps children associate the larger numbers with larger size.
Smooth, flat stones aren't just for skipping -- they're perfect for game pieces too. Making these dominos is an idea courtesy of Martha Stewart. Gather 28 rocks and create the dots and lines with a paint pen. Begin by drawing a line across the center of each. Then on either side of the line, mark with two sets of dots in every combination from zero to six.
Classification is the basis for all our choices and observations. Classification is the way in which children begin to make sense of their world; this is mommy, this is food, this is a car. Believe it or not, sorting and classification are pre-numeral math concepts with which children need lots of experimentation and discussion. And they can begin to do this at age 2! As a parent, it’s boring, and it’s tedious. But sorting and classification supports early numeracy concepts. It's so obvious that I can’t believe that I didn’t make the connection.
I am sure that Albert Einstein did not have a shape sorter. Yet “all the kids” have one, so Max needed to have one. Don’t get me wrong, shape sorters are fine. What I am saying is that I didn’t need to spend $20.00 dollars! Further, this should not have been his first “math game”. And I certainly didn’t need to stress about it when Max had trouble doing it! Einstein was pretty good at math, yet he probably played with a pile of old buttons or wooden teeth.
Almost any group of things you have in your home can be sorted in some way: keys, plastic animals, socks, dolls, cars, blocks, fake food. Chances are that you have some of these, if not all, scattered on the floor of your playroom. When it’s time to clean up, simply say: “Let’s sort these into piles as fast as we can! Let’s put the dolls here, and the food there. Go!”
While putting away the groceries, let your child “help” sort them by putting them into piles of cans vs. boxes, or refrigerator vs. pantry. Or simply sort laundry by color or by family member. Including sorting in every day activity and routine gives your child a “real” way to make sense of their world.
Playing tea party is a boring, but classic activity. Now that I am thinking about sets, this pretend game is taking on a new, and more interesting, turn. Now I’m asking Max to help get ready for the party and “set” the table. I make a set: napkin, plate, cup, and silverware, and ask him to repeat that set for each person. Sometimes he will count out the napkins, knives, forks, spoons, and plates checking that he has just enough of each before he takes them to the table. Other times, he sets one piece at a time. It’s fun to watch him develop his own strategy.
If you want to have an actual sorting “activity”, any spare box or bin will do. Try a muffin tin or an egg carton. They work well. Simply grab some random stuff from your junk drawer (pennies, buttons, nuts, bottle caps, paper clips, rubber bands) and ask your child to “help” you by sorting the pile into the cups.
As your child gets better at this game, you can introduce sorting the same objects in other ways. For instance, buttons can be sorted by size, color, number of holes, then by shiny or dull, etc. or on two traits. “Which buttons are both round and red?” I wasn’t surprised at all when Max came up with different sorting rules. And that was just fine. The important thing was for him to be able to find the relationships. In math, classifying similar objects is also known as creating a set. At 4 Max could classify objects by one attribute (size, color, height, length, or shape), whereas at 5 he has begun sorting by more than one attribute (yellow triangles or blue squares).
An interesting sorting game to play was described in Mathematics Their Way by Mary Baratta-Lorton. She calls it “What am I thinking?” This time I make the novel sorting rule. Again, my junk drawer and toy bin provided lots of similar items that could be sorted in many ways. Pick up one item from the pile. Say, "This can go here" and put it in one pile. Choose a another and say, "This can't go here" and put it on the other. Sort a few more, verbalizing, "this can, this can't...", then ask, "What am I thinking?" The idea is for your child to try and guess the sorting rule.
The next logical step in this is one of my favorite games: Eye Spy”. Max and I usually play this game while taking a walk or waiting for the waitress. Basically, one person gives clues to the other; “I spy with my little eye something that is red and tells cars what to do”. The answer is a stop light or a stop sign.
When Max turned 4, and had learned more about the world, we began playing “duck, duck, goose” with categories. Instead of saying “duck” the ducker names things in one category (colors or numbers or letters or kinds of birds) then instead of saying “goose” the ducker switches categories (sports, food, animals). All the kids get involved in the categories and it makes a dull game way more interesting, at least for me.
In math, sequencing is arranging a set of things, usually numbers, in order. The classic game of nested boxes is found in almost every 2 year old’s home. I must confess that I never saw the point of those boxes. But, they were cute so I bought them. We’d stack them up and Max would knock them over. The boxes do teach ordering but, neither of us found touching them to be interesting or satisfying in any way. What Max did enjoy was playing with all my empty plastic food containers, pots, and measuring cups. He’d stay busy for a long time touching and banging objects, learning shapes, size, and relationships.
If I had to do it over again, I would have encouraged him to stack them on top of or inside of each other. I would have challenged him to “line them up” in order from smallest to largest, lightest to heaviest, etc. Ordering is an important skill required to arrange numbers in a set as part of a math problem in later years.
Now that he is older, and pots and pans don’t have quite the same appeal, we arrange other “manipulatives”. Restaurants always give us crayons. I take a handful of used crayons and spread them on a table. I ask Max to arrange the crayons from smallest to largest. If they give us a lot of crayons, we organize them by color, and then within each color group we line them up smallest to largest.
By the way, don’t forget to employ music. There are loads of songs that employ counting and ordering: “5 Little Monkey Jumping on the Bed” is a classic. Plus, there are some terrific stories that support the notion of ordering by size: The Billy Goats Gruff and Goldilocks and the Three Bears are 2 of my favorites!
Last summer, while I still thought that counting was math, one of Max’s teachers was teaching patterns. It’s not like stripes or polka dots are any big deal. That’s girl stuff. Why should my SON know how to make patterns? I began to wonder is there some kind of bigger educational reason that I don’t understand? Perhaps because math was not my thing, I didn’t make some pretty fundamental connections. And it’s my research into this that inspired this article about preschool math.
I now know that patterning is a basic math. Patterns are not just the stripes on my shirt, rather,
patterns are relationships. Patterns are found in daily routine, weekly routine, and change of seasons. Patterns are found in music and art as well as in aspects of math such as counting and geometry. Times tables, addition and skip counting all require an understanding of and proficiency in patterning. Understanding patterns is based in sorting and categorizing.
I was surprised to learn that patterning is categorized under "Algebra". Really? Algebra? My kids is only 5! But, if you think about it, algebra is about seeing relationships, and as kids recognize and create patterns they begin to see and understand how things work together.
So as the parent of a 5 year old, what can I do to teach patterns? In a panic, I bought a bunch of sequence cards, pattern cards and attribute blocks and all sorts of educational toys that would “help” Max to see relationships and learn to create patterns. I spent a lot of money and the stuff took up a lot of space. I got really focused and pushed Max very hard. He hated it. He got angry. I got angry. I got sad. No one did patterns and the rest of the exercise (and day) was pretty much ruined. A few days later I tried again. Same result, except that Max began to feel that he couldn’t do it, and I got sadder. The “educational” toys got placed on a shelf, where they still sit unused and take up space in my small Manhattan apartment.
I gave up trying for a while and then started letting Max watch “Team UmiZumi” because they had a bit on patterns that was fun for Max to watch. Plus, he enjoyed shouting out the answers; correctly, I might add.
I have come to the conclusion that teaching Max preschool math concepts requires something other than handing him math workbooks or educational games. Kids need to have math experiences that are real and have purpose. In other words, it needs to be interesting and be “real”. I have figured out that for Max, using math tools makes math somehow separate from reality. That is not the message that I want to send. Yes, I want him to learn math, but not because math is interesting per se. I want him to learn math because it is part of life and living. If I teach it that way, it will resonate and interest him. How do I do this and where do I begin?
Any parent who has read any current parenting book, knows the importance of routine to a child. It helps children know what they are expected to do and to understand what will come next. Guess what? Routines are patterns! I never made the connection between the words “pattern” and “routine” and “sequence” before. In many ways they are the same. As of a year ago, I never thought about patterns as they relate to math. Perhaps if I had been shown the patterns that numbers make, my artistic mind would have wrapped itself around math!
Now that I’ve realized that patterns are everywhere and in everything, the task of teaching patterns is easier. Because I know that Max needs to experiment and make observations, and to have the material be interesting this will increase the odds that I’ll point out something interesting enough to hold his attention. I just need to take the time to identify patterns in our world.
We have blocks and we have legos….I suspect that most people do. Until now, we built with them. So, now, instead of just building, I set up patterns by color or by shape and see if he can extend them. As he does this, I try to have him read the pattern to me: red square, blue triangle, red square…. So far, he’s bored. But I am hoping that as he gets more proficient, he’ll enjoy seeing how long he can make the pattern repeat. What he does like is telling me what is missing, if part of a pattern is hidden, or where I have made a mistake. The important thing is that I have begun to get him thinking about color, shape, size, and relationship.
Max showed me that this catepillar had stripes. Something is clicking!
So, I began to discuss patterns in our daily routine, days of the week, and seasons. We also made it a daily habit to discuss order: what did we do first, second, last? I’ve started to show him patterns on buildings, and he really thinks it’s cool that the crosswalks have a stripe pattern in them. But it still hasn’t helped him to create a pattern.
Then it occurred to me, patterns can be visual, physical, or auditory. We have begun to create patterns with our bodies! Hop on your left foot twice, and on your right foot once, or clap in a “rhythmic” pattern. Suddenly I don’t find “Head, shoulders, knees and toes…” quite so dull!
As I said earlier, I did a lot of counting with Max. He became a fairly good counter early. However, there is a big difference between counting, and counting something. As early as 2 years of age, many children will parrot the words 'one', 'two', 'three', 'four', 'five' etc. However, rarely do they understand that the number refers to an item or a set of items. This is known as 'number conservation' or 'one to one correspondence'.
For a child to put their finger on something and say one, then put their finger on the next things and say two is more important than just being able to count by rote. I did not understand this difference, or maybe I just took the meaning behind counting for granted. Again, I should have had Max count real objects, and point to the object as we counted each one. After counting, I should have repeated the total with the item name: “1, 2, 3, 4, 5 we have 5 raisins”. This points out that counting lets them know how many things there are in a group.
Fingers are tools you always have with you; use fingers to count. Put up a finger one at a time as you count it. If you’d like to have an actual activity, draw a number of circles (faces) and put down a number of buttons for eyes. Ask the child if there are enough eyes for the faces and how they can find out. Repeat this activity for mouths, noses etc. Speak in terms of more than and less than or as many as and how can we find out.
I had always heard that music was linked with math and dutifully brought Max to Dream Jam World classes, patting his little body along with the beat. Now that I know more about early numeracy concepts, the patting they recommended makes sense. As Max grows, I now see him marking the beat (pattern) of the music while stomping or clapping. He is matching his stomps one-to-one with the beat. So cool!
Counting and one to one correspondence leads to a topic called “number sense”. Number sense is much more than counting, it involves the ability to think and work with numbers easily and to understand their uses and relationships. Number sense is the ability to count accurately and competently, to be able to count on from a specific number as well as to count backwards, to see relationships between numbers. It is about counting, adding, and subtracting.
So, back to the counting that I did with Max. At least I did this right. Counting is good exercise to foster number sense. People way smarter than I, recommend that parents help their kids practice counting. So, count everywhere you go. Count forwards, and backwards!
Preschool teachers say that parents should count real things to help children use their own experience with objects to better understand numbers. So now Max and I count our steps, telephone poles, houses, cats, trees, etc. I have begun to talk to Max about what numbers are used for: keeping score in a game, prices, or finding a street address. When we go shopping, I show him where to find the price of items and then I read the amounts out loud. Some numbers, like those on baseball uniforms, are used like names; others are used to tell you the order of something, or the amount.
I write numbers on everything and we play games like: think of a number between 1 and 10. I give clues like "bigger" or "smaller" and ask him to guess the number. Besides being a fun way to pass the journey, it helps him develop a "mental number line" as he thinks about different numbers and how they relate to one another.
You will need to do one to one matching activities to develop “conservation of number”. Children need to 'match sets' before they will understand 'number conservation'
So now, this is how we eat goldfish crackers. Instead of just eating and eating, I tell Max that I’m going to tell him a story while he eats, but he has to be patient and only eat when the story tells him to. I then proceed to weave a tale about a little tuna fish that roamed the seas with lots of huge fish. "He wanted to be bigger, so he ate 3 goldfish (cue to eat 3 crackers). That did help him grow, but he wanted to be even bigger, so he ate 5 goldfish," and so on. You can structure the story any way you like; sometimes the protagonist is a tuna, sometimes a fisherman or an octopus, or sometimes the goldfish escape down the drain to the ocean. After each swallow, I ask Max how many goldfish are left, and help him count. Gradually, he’s developing an understanding of what numbers mean, how they are used, and how numbers relate to each other.
At 5 Max can relate to simple concepts of addition or subtraction. For example, if I ask “You have four blocks. If I take away one, how many will you have?” I am finding that he can add small sums in his head, but he still can’t do it just using numerals.
Conservation is an intersting aspect of quantity that falls within number sense. Conservation of number is the understanding that the number of objects remains the same even when they are rearranged spatially. We all know that children are guided by their perceptions. Given 2 piles of 6 things each, they will think that there are more items in a pile of larger things. For instance, volume-wise, a pile of 3 grapes is bigger than a pile of 3 raisins. Most preschool children will conclude that there are more grapes! Another example of this is that if you have 5 items next to each other, and another set of identical items spread apart, the child might think that there are more items in the set that is spread apart!
Again, you can use food to teach math! Make a pile of 5 grapes and another of 5 raisins. Ask your child to move one raisin and you can move one grape. Repeat the process so that they can see the number is the same. These experiences will need to be repeated often. You can also try this one: arrange items on a tray (toothbrush, comb, spoon etc.) ask the child to look away, rearrange the items to see if they realize the number of items is still the same or if they think it's different. You’ll learn a lot about your child’s development if you do this every once in a while.
Charlotte is definitely IN the box!
It’s easy to make your own puzzles from catalogues by cutting up a picture into large pieces. You can add numerals for older kids. And if you want to make a more permanent puzzle, simple glue the photo to an old cereal box before you cut the pieces out. This puzzle took about 8 seconds to make and we played all sorts of games with it for about 20 minutes.
I’ve also made puzzles out of popsicle sticks. If Max likes to draw a little more, I’d have him draw the picture and then play with the puzzle.
If you need an actual activity, collect the covers of damaged circular plastic covers to make a pie fraction manipulative. These covers come in standard sizes, and have the center of the circular covers marked by the mold they were made with. Simply cut them up into the different fractions required. You can also make fraction strips from old book covers, or plastic folders, simply by cutting them into strips of the same size, folding them into different denominators and marking along the folded edges.
The bathtub is a great starting point, using a variety of plastic cylinders/cups and containers. Practicing measurement, your child will learn how big or little things are and how to figure that out. I have given Max a variety of objects to take into the tub and asked him to predict whether it will sink or float. Try soap, a ping pong ball, a toy action figure, or just about any waterproof object.
We have started keeping a record of everyone’s height by marking the wall and measuring the height every so often. He loves seeing how big everyone is getting and predicting how big he will be next year. The marks on the wall are simply a graph of their growth. Don’t get me started on graphing, we’re not really up to that!
Geometry is the area of mathematics that involves shape, size, space, position, direction, and movement, and describes the physical world in which we live. I always loved it! I loved putting pieces and puzzles together. I loved coloring in graph paper to make complicated designs. Don’t get me wrong, I didn’t enjoy trig. or calculus but geometry and shapes still resonate in my mind.
Geometry and spatial sense help children with directions and finding their way around. It’s important to let them climb in and out of boxes, on or around furniture, going under, over, around, through, into, on top of, and out of different things to experience themselves in space. Spatial sense gives children an awareness of themselves in relation to the people and objects around them. While Max did this, I used to sit and watch. Other than turning it into a round of Hide and Seek, I wasn’t quite sure how to join in. Now I see that I could reinforce his play by giving him the vocabulary to foster his understanding of space: “The cat is under the bed,” “you are between the wall and the door.” You’re teaching math every time you talk about locational concepts like near, far, left and right. When we do this, I’m working on those Einstein skills.
Children learn geometry best through hands-on experiences: blocks, boxes or containers, and puzzles. But, my mom taught me geometry with food. Sandwiches are a great medium for introducing beginning shapes. As my mom did, I always ask Max if he wants his sandwich cut into 2 rectangles, 2 triangles, or 4 squares. I needed to demonstrate these shapes a few times but now he’s a whiz at shapes. Through this “food play” Max is also learning about the relationship between parts and wholes and about beginning fractions!
Young children learn about angles, shapes, and solids by looking at the physical world. One of Max’s teachers told me to have him identify and describe different shapes, to draw them in the air with their finger, to trace over them with their fingers, and to draw them on paper. She suggested that we go on a “shape hunt.” When Max and I go to the grocery store, I use it as an opportunity to find shapes: circles on the can of soup or the square on the box of rice. It helps him begin to recognize, identify, and describe shapes. This is geometry.
Identifying shapes lays the foundation for those geometry lessons in the future. When Max proudly showed me the triangle he had learned to form using his thumbs and index fingers, I imagined the geodesic domes and flying buttresses he’d one day employ as an architect.
When Max was little, I would set up a mystery game in which he could feel shapes and then figure out what it is. I just gathered pairs of things. I put one of each into a bag and one of each on the table. Max would reach in and feel the shape and guess what it was.
Fractions represent parts of a whole. A very young child will see something cut into three pieces and will believe that there is more after cutting it than before it was cut. To understand fractions, children need to think about: what the whole unit is, how many pieces are in the unit, and if the pieces are the same size. When cutting an apple into four equally-sized pieces, say, “I’m cutting this apple into fourths” or at lunch, announce, “I’m cutting your sandwich in half.” This connects the language for fractions with the concept.
Many sharing activities help children understand fractions by dividing food, chores, or treats into equal portions. Cutting up a pizza is a good way to get children thinking about fractions. Share a banana, sandwich, or cookie by splitting it into two parts of the same size. I explain to Max that each of you is eating one-half. He is beginning to see that as more people share, the size of his share gets smaller. I am also encouraging him to solve simple problems. “There are nine crackers and three children. How can we make sure everyone gets the same number of crackers?”
Just because preschoolers can’t use measuring tapes or scales yet, doesn’t mean that they can’t measure things. Quite the contrary. I watch Max and his friends measure things all the time! Usually, they employ direct comparisons instead of rulers. For example, they may see a dog in the park, and know that it is smaller than the dog at home. Max and his friends enjoy saying that they are 'bigger' than their sister or brother or 'taller' than the lamp or that they are 'higher' than the dishwasher.
At this age, perception is Max’s guide, he does not have any other strategies to help him in determining which has more or less, is heavier or lighter etc. I have actually heard him say that he has “more” in his cup simply because it’s taller. Before age seven, children will not know that the amount of liquid in a short, fat cup remains the same even if the liquid is poured into a tall, skinny cup. They will think that the taller cup has “more” in it because it looks like more. The best way to guide understanding of measurement is through measuring activites.
When direct comparison isn't possible, such as when figuring out which table is longer when they're on different sides of the room, children can use a variety of nonstandard measures -- paper clips, pencils, baby steps. Encourage them to “measure” objects using non-standard units. For example, ask, “How many blocks long is the (TV remote/coffee table/couch)?” Making direct comparisons and using nonstandard measures help prepare children for learning standard units such as inches, centimeters, and feet. Measurement is an important way for young children to look for relationships in the real world.
Many daily activities involve measurement: cooking, gardening, grocery shopping, sewing are only a few examples. Max loves to be involved in these chores and we chat away about what we are doing. When I have something to measure, I let him help by holding the ruler or the yardstick.
As Max gets more familiar with the concepts of numbers, quantity, volume and measurement, I will expect that he could begin to make educated guesses or “estimates” about the amount or size of something. Very young children will not be able to estimate accurately, because they are still learning these concepts. When children use estimation, they learn to make appropriate predictions, to obtain reasonable results, and they learn math vocabulary such as about, more than, less than, near, approximately, in between, and around.
Estimation activities are easy to create. Just take a guess about something—like which one of his friends is the tallest—and then check it out for accuracy. To help Max learn estimation, I write down the estimate and then the actual count. If I repeatedly give a similar problem, Max will eventually estimate closer to the real count. It isn't important to get the "right" answer, but to see how close he can come.
Estimating can be another fun food activity. Set out 2-3 bowls filled with small amounts of food, such as; 8 baby carrots, 12 grapes, or 10 fish crackers. Before eating their snack, have your child try to guess (estimate) how many items are in their baggie. Next, have them empty out their bag and count their snacks.
Another interesting challenge, for older kids, is to draw a line on a piece of paper and ask them to guess the fewest number of crayons (of any size they choose) that will equal the length of the line when the crayons are laid end to end. Then have them actually line up the crayons and see how close they came to the right answer.
Children learn by thinking for themselves: let them make their own estimates and then check out their accuracy.
We enter the world as mathematicians, exploring all the ways we can order our world, craving an understanding of the logic of things. Kids are designed to try to make sense of their world. Turn it on and turn it off and turn it on again. Pick it up and put it down and pick it up again. Put it in and take it out and put it in again. It might drive us crazy as adults, but really, the children are simply testing their formula, practicing it until it's second nature: A-B-A-B-A-B . . .
Through play, children go on to discover increasingly complex patterns all around them. Then they use those discoveries to do important things like take turns or draw stripes on a tiger. Kids are mathematicians discovering for themselves the classifications and patterns of the world.
I can't speak for college level math, but I know that there is no reason that math shouldn't be fun. Math is not hard, but at some point, for many of us, it stops being about discovery, which makes it no fun. If you think about it, rolling down a hill is a physics exercise and mixing mud is a lesson in proportions. Math is in nearly everything our children do. They are programmed to learn math and we are programmed to teach it. Forget the work sheets, homework and memorization. Math is supposed to be fun.
The list of ideas on how to include math into the daily living with a preschooler is endless and the list of ways to make manipulatives from recycled and found materials can run into tens of pages. Learning preschool math should be so much a part of regular home activity that your child won’t even realize that he’s learning math! All it takes is a parent who is determined to spread the word about green living and a love for mathmatics.
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