Goal: To learn about everyday probability

Materials: Manipulatives, such as unifix cubes, basket to draw the manipulatives from

How to play: When Penny finishes her meal she gets desert. How much dessert she gets, however, depends on what color of cube she selects! We put one blue cube, one black cube, and eight yellow cubes into a basket. She then closes her eyes and draws for desert. A yellow cube equals one dessert (one sucker, for instance) a blue cube equals double the dessert, and a black cube equals – gulp! – no dessert.

I like this game because it works math into an everyday situation: dessert time. My hope is that Penny learns, first, the idea of what it means to say that something is more or less likely than some other event. My second hope is that the language of probability (likely, chance, probability) starts to mean something more to her than just some word she might have a vague notion of. Most of the time before she draws I have her go to the whiteboard and she will write down the fraction that represents the chance she will draw a certain color (1/10 for a blue cube, for example). She is still learning how that written fraction relates to her chances of drawing double desert, but we’re getting there. The thrill of drawing makes the whole desert-time activity fun and exciting. The one time she has drawn a black cube, I ended up giving her some dessert, since she did eat all of her meal. No need to make desert time traumatic!

After Penny has gotten a little more facile with the idea of this probability game, I will start to track what she draws. Then we can look at the chart or graph that shows that, indeed, the yellow cubes are drawn more often than the blue or black cubes. We can then relate that back to the Math Mountain game (where we recorded dice rolls and graphed the result) and see if we can spot any common themes between the two.

Certainly the intermediate goal of all this is to help Penny have that Eureka! moment, when she applies what she’s learning at home to something she sees while we’re out and about. This will show that she can apply the information, rather than simply regurgitate what I’m telling her, and will also show that she sees how these math games relate to a much broader world out there.

As an aside, the other day I was out with Penny and we were talking about some things we noticed in a Target store. When I was about mid-sentence, she blurted out, “that’s just probability you’re talking about.” A passer-by couldn’t help but stop and ask, “did that little girl just say “probability?”

Well, yes, she did, but my response would be that she’s not any more or less smart than your children. I just choose to spend a lot of time with her talking about these type of topics. It’s not hot housing, it’s not stultifying, and it’s not damaging in any way. Rather, like all children her age, Penny is a sponge, is driven to learn new information, and has an intense curiosity about nearly everything. Instead of filling her head with the latest Michael Jackson funeral proceedings or People magazine, why not fill her head with great books, deep concepts and grand ideas? Why not sow these seeds now, so that they sprout as she starts Kindergarten this fall and throughout her life in school and beyond. If she wrestles at this age with tough problems, deep issues and curious facts, those same things will be second nature to her when she’s called upon to bring her ideas to the table.

Goal: to learn about dice, number composition, and probability, and to practice addition.

Materials: two dice, pencil and paper

How to play: along the bottom of a piece of paper write the numbers 2-12, so that each number can form its own column. Roll a pair of dice. Put an ‘X’ above the number the two dice add up to. So, if you roll a 3 and a 4, put an X in the 7 column.

The general idea behind this game is to serve as an easy introduction to probability, though there’s little overt discussion of this when we do this. As we rolled the dice enough times, we started to see a pattern emerge, one that looked like a mountain that towered to the clouds around the 6, 7 and 8, and rolled off into the plains near the 2, 3, 11, and 12. Hmm, we wondered, why does this happen? I briefly explained that there’s only one way to roll a 2 or a 12, but several ways to roll a 7. Penny likely doesn’t fully understand this yet, but she has a basic understanding of what’s happening, and the practice of adding the two dice up is just gravy on top of, well, on top of the mountain.

As she gets older and more comfortable with math, I plan on going much deeper into the math of dice (probablity) and the math of many of the other games we’ve played. Hopefully she’ll be well-prepared given the base she’s getting, and will be able to draw on this knowledge to make connections and reason deeply on her own.

Goal: to learn about number composition, and to help with addition and subtraction fluidity.

Materials: pencil and paper, interlocking cubes

How to play: pick a number of cubes appropriate for your child’s age and ability (we used seven cubes tonight, for example). Have the child arrange the cubes into some pattern. For instance, lock two cubes together, then two cubes under that, then a single cube, then finish off with the remaining two cubes. Reading from the top down, along the rows, that is, say out loud the pattern you created: 2 + 2 + 1 + 2 = 7. Then have the child write this equation down. Then have her rearrange the cubes into a different pattern.

I picked this game/activity up from reading a book at Barnes and Noble tonight. Sadly, I cannot recall the title, but this game stuck with me. I’ve been looking for ways to help Penny understand what is called number composition, which is the various ways in which a number can be constructed. Using the interlocking cubes lends a very physical way for her to understand this concept, and it adds a level of entertainment for her, beyond, say, using dots on a page.

Kids generally like building and tearing things down, so this is a very natural way to help inculcate this concept. What’s great is when she wants to quiz me about the games we’re playing, as I do her. So she quickly wanted to start to ask me what the pattern was, after she had created a new pattern. I think this “teaching mode” is a great way for her to start using the language of math, as she does her best to mimic my style (poor girl) and speaking patterns. To me this just underscores that I must pay attention to phrase things correctly, so that her math verbal habits are correct from the start. You likely already realize that kids pick up a lot more than we think they do!

This isn’t a math game, though I decided to mention it here. Tonight I decided to introduce the idea of digits to Penny. Not sure where I’m going with this, but in general I think teaching her the language of mathematics is important, and also allows me to make up different games involving the number of digits in a number.

We first learned how to find the number of digits, which is simply counting the numbers…in a number! So 24 has two digits, and 300 has three digits. I then showed her some crazy! things, like how two 1-digit numbers can sometimes add up to a single 2-digit number! (5 + 5 = 10). We had fun with this, and I quizzed her later in bed and she got tickles if she got a wrong answer. Hey, if she’s laughing about math, I’m doing my job.

I don’t want to push it quite yet, but I feel it shoudn’t be more than a few months before I can start showing her how to add multi-digit numbers on paper, using the carrying method. I don’t feel she’s ready for that yet. Subtraction will be much later, I feel. It might be a situation where I introduce it and then wait for her to show interest in how I did it.

Anyway, I’ll be thinking of some game we can play using digits. If you have any ideas, please let me know!

Goal: to accurately tell a story using the language of math.

Materials: paper and pencil, perhaps manipulatives

How To Play: tell a short story, perhaps as short as a sentence or two, and have your child write down the story using math. For example, “I had three apples, but Penny was hungry so I gave her one. How many apples do I have left?” The child should write the equation, or math sentence, or “story,” as 3-1=2.

I love this game. It can help a child make that jump from mere rote pushing around of manipulatives to an understanding of how math is used daily. The child is introduced to many different ways of saying ‘added to” and “subtracted from” by the varying language used in the stories. In working with Penny and doing these math stories for perhaps three or so days now, I have already learned that for her it’s best to adjust the difficulty of the story by adjusting the language I use, all the while keeping in mind her level of interest, and her most recent experiences. Quite naturally she feels more confident in her ability to solve a problem if she has had recent success, so I keep that in mind when telling more involved stories, or making the language a bit trickier. Here are some examples:

1) We had a full Saturday. We went to three different places. We were ready to go home, but Penny wanted to go to two more places, so we did. How many places did we go altogether? Can you write that story down in math?

2) Penny had seven chips. Daddy was VERY hungry and so he took all but two of them. How many chips did Daddy take? Can you write that math sentence down? (Please note that we are not yet at this level!).

3) Penny has four stickers on her arm. Everytime she goes to Mimi’s house, she ends up with two more stickers. If Penny goes to Mimi’s house today, how many stickers will she have? Can you…

4) Daddy has one book. Penny wanted to read a book, so Daddy gave his book to Penny. How many books does Daddy have now? Can you…

My goal is to help Penny learn the different language used everyday in expressing mathematical ideas, and to help her become comfortable with seeing math as a language, one that can tell a story and eventually describe the world. A more mundane, but still practical, goal is to help prepare her for that bane of most math students: word problems. I figure that if she starts early and internalizes just what an equation is “saying,” then it’ll be that much easier for her to both read and write “in math.” I also believe that this type of activity will help her to see that math is not merely pushing around cubes to find out what some number added to some number is. Tying numbers and math to everyday activities will hopefully serve as a motivation to want to learn more advanced math so that she can model more advanced activities.

Goal: to make a delicious math soup!

Materials: multi-colored manipulatives, bowl

How to play: write down a simple recipe for your child to follow. For example, my recipe was 4 reds, 4 white and 7 blue chips. Your child then reads the recipe and cooks up the soup for you to enjoy.

This was a very spontaneous game. We had just finished playing Math Trade, her favorite game which I discussed in an earlier post, and we had all of our poker chips lying on the table. She wasn’t quite done playing so Penny announced that she was going to cook some supper. She also announced that she was going to write down some recipes to follow. I just thought it would be neat to have her follow a “math” recipe to make me a delicious dish. She was a bit confused at first, but she was able to read the entire recipe and the feast was enjoyed by all.

At this stage the astute reader will notice that there isn’t much math involved in this repast at all. Well, this time I just put a simple number down for her to follow, as we had just finished two straight Math Trades, and so didn’t want to get burned out. This game is easy to modify, however, to include all sorts of math-related skills; for instance, each recipe could have been a simple addition or subtraction equation for her to solve. The beauty of a lot of the simple games that can be played between child and parent is that they can be make more difficult or more simple, easier or harder, quite simply. This allows you to grow with your child as she grows more adept in the basic math skills, and more confident in her ability to tackle areas just beyond her understanding. This is a key point as my experience has taught me that Penny’s willingness to attempt a novel math concept in large part is contingent upon her recent level of success in any of the math games we’ve recently played. More recent success leads to greater willingness to accept challenges. Thus the games that allow for modification allow for more instances of possible usage. It might sound technical, but the parent is typically good at reading their child, and cranking up or backing off at the right time can go a long ways toward keeping your child’s interest piqued and her learning on track.

Goal: to be the last one to play the game piece

Materials: any manipulatives, such as poker chips

How to play: we start with 10 chips. The goal is to move pieces from the start pile to the finish pile, trying to arrange it so you are the one to move the last piece from the start pile. Player 1 can select either one or two chips, and move them to another pile. Player 2 can then select either one or two chips and move them to the other pile, and Player 1 goes again.

As with many of the other games I’ve written about here, this one is fun not only for very young children, but can also satisfy a more advanced player. While the rules are easy to learn for even a very young child, the theory behind winning can touch on some relatively advanced topics. With a new emphasis on patterns, I talked with Penny about any patterns she discerned in our play. I don’t blame her for not noticing many (she just wants to beat old Dad!), so I did talk with her about the idea of winning, which is forcing your opponent to have the wrong number of chips when it’s her turn…the trick is how far toward the beginning do you start this planning. The strategy hinges on how many chips there are at the start, and even how many each player can take each turn (1 or 2 is only a suggestion).

I reiterate here that I believe these types of games are more useful to a young child than more rote activities such as working out a flash card addition problem by counting out manipulatives, then counting out some more and seeing how many you have. While the rote activities are useful for teaching a certain technique or algorithm I think a very young child, which I would define as one not yet in Kindergarten, gets more from these games where they spend some time thinking about strategies, dealing with the consequences of their decisions, and competing for fun with a goal in mind.

My daughter started her pre-K class a few weeks ago and a comment by the teacher caused me to, in some sense, rethink my strategy in working with Penny on math-related items. Her teacher said that she would be stressing the study of patterns with her children. It immediately occurred to me that patterns was something I didn’t stress nearly enough. Patterns are such a large part of math that many people define mathematics as “the study of patterns.” So over the last few weeks we’ve spent a lot of time doing what I call Pattern Play.

We do Pattern Play in many different ways and formats, but a mainstay is by using small, interlocking colored cubes, which can be found in teacher supply stores and most toy stores. My wife or I will make a pattern and have Penny copy it, or make some random patterns and put them alongside repeating patterns and ask her to tell us which of the blocks form repeating patterns. We’ll let her make a pattern which we’ll repeat, or maybe I’ll ask her to make a pattern with four elements in it; that is, a pattern that repeats after the fourth element.

A favorite of Penny’s is to make a pattern of ascending stairs using the cubes. She’ll start with one, then with two and so on until all of the cubes have been used up. As she gets older there’s a lot of math that can be explored with something as simple as this. I believe she will appreciate the different levels of complexity that can be drawn from something she was able to enjoy as early as pre-K! We’ll also do stairs using a two-cube step up, or even three. She’s practicing her rote counting, small-object manipulation, pattern visualization and addition/subtraction skills.

Just yesterday we also went on a nature walk and looked for patterns we could see in nature. Penny is very interested these days in writing, spelling and drawing so we brought along with us a “science journal” and recorded and drew some of the patterns we saw. I believe I will purchase a special notebook that we’ll use as her science journal. How neat for her would it be many years from now to look back and see her science notebook from even before her Kindergarten years?

We also study patterns in number charts. I copy a 1-100 number chart and we color all of the numbers that have a five in them, or a seven. This will form a “plus” sign, and it’s an interesting way to see how the fives are distributed throughout the number chart. We’ll also color in the odd or even numbers, or we’ll count by fives and color only those squares. This again is one of those subjects that can grow along with her, as much of number theory is based around seeing patterns in whole numbers. I’ve got books lining my library wall about number theory and if she should by random pick one out, the games she played with number charts will hopefully become a stepping stone for understanding and then enjoying that book. Or at least that’s the idea!

The more I think about math the more I realize the importance of patterns. My goal in directing more time toward Pattern Play at the expense of addition and subtraction using manipulatives is that being able to detect, create and understand patterns is the more important component of math in the long run. Penny will learn the addition and subtraction algorithms just fine, but being immersed in patterns, having them called to her attention and thus hopefully seeing them more and more around her because of it, should give her a head start in math both now and in the future.

Goal: to arrange scrambled playing cards in the correct order

Materials: playing cards, with face cards and joker removed

How to play: arrange 10 playing cards in a line on a table out of order. Tell the child that the snake is broken and that to be fixed he has to be put in the correct order.

This very simple game helps children become more familiar with the ordinality of the numbers from 1 to 10. Rearranging the cards helps the child think logically. This game can be extended to using dominoes, so that some addition is required to find the value of each domino. Likewise flash cards can be used, either subtraction or addition. The latter idea is quite a bit more difficult for a pre-K to Kindergarten child as they have to do quite a bit of either addition or subtraction to see what each card is worth and those operations at this age are still a bit cumbersome. A child this age would likely need some direction and assistance to understand exactly what she needs to do to fix the snake. Keeping her attention through too lengthy operations exercises can also be a challenge, and often a strong sense of “who is going to win!” is needed to keep her interest piqued when there is much perceived drudgery involved in arriving at an answer. I believe that as she develops more of a sense of addition’s and subtraction’s place in everyday living she will be more tolerant of the process of going through these operations; it’s important at this age, I am finding, to keep multiple steps in any math game closely linked and directly related to the ultimate task. Helping her to see the multi-connectedness of each step will eventually help her to put each step in the right place and help her to understand how everything fits and reasons behind why any one such operation is used.

Goal: to be the first to score 10 points

Materials: poker chips or other manipulatives, cup

How to play: arrange a small number of chips, or “mice,” on a table, probably 10 or less for a child under five. Make sure the child sees how many chips are on the table and then ask her to close her eyes. While her eyes are closed, slip some of the chips under the cup. Then ask the child to open her eyes and try to guess how many of the “mice” are under the cup. Correct answers get a point!

I tried this game for the first time with Penny this morning after getting the idea from Chris Confer’s book Teaching Number Sense: Kindergarten. Chris’ book is a great resource for ideas, theories and application of games and methodologies for teaching math to children. She describes a game and then relates her experience playing this game or teaching a certain concept to Kindergarten level children, complete with their quoted thought processess, successes and failures. She also has a 1st-grade level book which follows the same format. If you find the information on this site helpful, then you’ll find her books doubly so. Books liker hers, and others, is what this site is at least loosely modeled after.

The best thing about this game is that it’s a great way to help a child to grow away from merely being able to solve a subtraction problem by simply following a mechanical algorithm which she hardly understands, to using logic to come to the conclusion Hey, here is where I can use that subtraction stuff Dad was talking about! Penny is quite capable of looking at a flash card and saying, Ok, I count out 10 chips, then I take away five, and that means I have five left! After this one fairly brief session, however, that next step in understanding is not yet with us.

My goal with the Hide The Mice game is to allow the opportunity for Penny to reason for herself that if we started with 10 mice, and now only five mice are showing, I can subtract five from 10 and compute that there are five mice hidden under the cup! I suspect that this is advanced thinking and reasoning for a 4 1/2 year-old child. I cannot corroborate that as I don’t spend much time with other children of her age, and hardly ever am I doing anything related to math with other children, nor do I have any formal training that would help me know. However Confer’s book does a great job of recording the whole conversation between a group of Kindergarten students and Confer herself as they go through a math game or concept and I would say that Penny would fall somewhere in the high-middle of that group for conceptual ideas, and near the top for mechanical activities, but she has almost exactly a full year before she’ll be in Kindergarten, so I am positive that this will be old hat for her by then.

In retrospect I feel that I should have let her fail a bit at this game. To explain, when she would attempt to figure out how many mice were under the cup, she would just guess a number, albeit a number less than the total we started with. I would then ask her why she thought there were that many mice and she would say “because there could be that many.” True, and I agreed with her, but then I should have let her peek under the cup, count the mice, and see that her guess was not correct. Rather, I talked her through the reasoning I would have gone through. Perhaps my biggest foible is not being able to always withhold teaching. In other  words, I would like to better resist the temptation to tell her how to go about figuring the answer, and just let her hit upon the answer herself. The danger with something this relatively complicated is that it could take her a real long time to reason through and come up with the right solution and frustration and disillusionment might very well come around before success. It’s an issue I’m struggling with and I’ve read a lot of different ideas from a lot of different people on what’s best and any consensus is nebulous.

Anyway this is a game I will revisit quite often until she’s not only completely comfortable with the mechanics involved, but also until she truly understands what is happening, why subtraction is the tool to use to figure the answer, and can show her understanding by applying this same concept to a related but different application. All in all it’s not the mechanics I’m so interested in her learning, though they are unavoidable and necessary. It’s the understanding and the fluidity of conceptual tool usage I’m going for. I think the Hide The Mice game is a great way to help this process. And who can resist finding hidden mice?