Teaching Probability to First Graders

Even though we're coming to the end of our two-week December vacation, I feel the need to write about the math unit we just finished. And why, you ask, should this be the subject of a post? I have some other tales to tell, and have done some thinking about how I'm going to change some routines once we get back to school next week. However, the words of our school Math Specialist have been echoing in my brain for days.

One day a couple of days before vacation I thanked Ms. B. our Math Specialist for her help and mumbled something about our having "done some really good math." She replied, "I used to do this lesson wth my seventh and eighth graders!" Her comment reminded me that ten years ago I had used an activity very similar to the one we had just completed with my sixth grade when we had worked on Data and Probability. I didn't have time to reflect on that surprising fact at the time, but it made me want to write about it here, it being the fact that we had just completed a very sophisticated math activity previously taught in Junior High to 1st Graders and... they understood it!

Our new math curriculum (ThinkMath!) has one unit on data and probability. We began our unit by learning how to organize data using tally marks. The students understood quickly the logic behind using tally marks. They saw how easy it was to count by fives, and add on to groups of five rather than counting by ones, where it's easy to lose track and have to go back to the beginning. We transferred our data from tally mark form to graph form, building bar graphs and, in a subsequent lesson, pictographs. We talked about what we learned from our graphs. The work in our "Lab Books" required us to compare the totals in our graphs to the totals in our tally tables. We compared totals and saw how much more one group had than another. Students worked eagerly and with excitement on their Lab Book pages.

As I look back, it is easy to see why my students enjoyed this work -- they could easily observe which column has more, whether it is more toy cars or more yellow or green balls. The math is fresh, accessible and fun! Students who find reading challenging easily make sense of graphs and answer questions surrounding the graphs using language with words more and less.

One aspect of our new math curriculum is the emphasis on language that we use to talk about mathematical ideas. After representing data and information about ourselves in graphs, we moved on to the ideas of certain, impossible, likely and not likely. I wrote the word impossible on the board and asked the students if they knew what it meant. We agreed that a good definition would be that "it can never happen." We talked about what it meant to say that something was likely. I mentioned that even though it has the word "like" in it, "likely" didn't mean anything about "liking." We said "likely" meant that "there was a good chance it could happen, but it didn't have to happen." I noted that when you used the letters (prefix) "un" it meant "not." "Not likely," or "unlikely" meant that "there wasn't a very good chance it would happen." "It probably won't happen, but ... it could happen." "Certain" we defined as "absolutely positively happening." No doubt about it, certain meant a "sure thing."

We had some fun talking about things that could happen to us that were certain, impossible, likely and unlikely. "It's likely that we will have outdoor recess today, because it's not raining." "It's unlikely that we will go to the beach today." "It's certain that school will be out today at 2:20." We made a joke that said something about it being certain that Mrs. D. would lose her glasses at least one time during the day, which everyone in our class knows is absolutely, positively a sure thing.

The next activity in our unit was to play the Number Race Game. This is a game played in pairs, with the two players sharing a pair of dice. Each player has a paper with the numbers from 0 through 12 in columns across the x axis or bottom of the paper. The y axis is the number of tosses of the dice with 8 being the highest number that can be achieved. The object of the game is to see which player reaches the top of the column first. The students take turns throwing the two dice, and then recording the sum by coloring in the correct column. For example, if I throw a five and a three, I color one of the boxes above the eight. There are eight boxes in each column. It takes surprisingly long -- almost 25 minutes -- before most of the students reach the top of the column. We gather on the rug with our papers to discuss what we've done.

I've made a chart on easel paper that lists the numbers 0 through 12 on the side and the words "how many wins" is a column heading. I ask the students "How many of you won with zero?" No one raises his hand at first, but then one boy does. "Jon, what do you think? Did you get to the top with zero?" "You can't get zero!" he says, "You can't get one either!" A bunch of students shake their heads and agree with him. "You can't get zero? Why not?" I ask everybody, feigning ignorance. Jon continues. "When you shake the dice, you have to get at least two because one is the smallest number you can roll." My heart beats a little quicker because I can see that the students understand this. They know that you can't roll zero on a die. I revert to our earlier language lesson. "So it's impossible to get a total of zero?" YES!!" the whole class shouts. "Is it likely I will roll a two? How do I get a sum of two?" Elizabeth raises her hand and answers, "you have to roll two ones to get two. I don't think it is very likely, but it is possible."

I asked each child what number they had won with and our chart ended up with one person wining with five, three winning with six, five winning with seven and six with eight. No one had won with two, three, four, nine, ten, eleven or twelve. I ended the lesson with the question, "Why do you think no one won with any of these other numbers? I think we'll play the game again and talk more about it tomorrow."

They had played the game and the game itself was fun. Coming to the rug to deconstruct the game was a little annoying to them, and they really didn't want to do it. They knew I was looking for something and we'd worked for so long with a discussion on words, then playing the game, then talking about it....I knew we had to go back to the words again, so I started the next lesson with a review of those four words: certain, impossible, likely and unlikely. Then we talked about dice.

We talked of the numbers on a die: one, two, three, four, five, six. I reminded them that yesterday they told me you couldn't get zero or one if you threw two dice. We talked about the combinations you could need to roll to make a sum of three? You'd get 1 = 2, then 2 + 1 or two combinations; I asked them how you could get the sum of four? we made three combinations: 1 + 3; 3 + 1; 2 + 2. What about five? 1 + 4; 4 + 1, 2 + 3; 3 + 2 -- four combinations. Six? 3 + 3, 1 + 5; 5 + 1; 2 + 4; 4 + 2 -- five combinations. One little girl said out loud, "I see a pattern. It's getting bigger!" Seven: 1 + 6; 6 + 1; 2 + 5; 5 + 2; 3 + 4; 4 + 3 = six combinations!

We then made eight: I began, "Okay, let's make eight! I'll add.... 1 + 7, 7 + 1...." Faces were raised to me, thinking hard, "She's right, one plus seven IS eight!" Melly raised her hand and couldn't wait. "You can't use seven!" Faces turned to her in confusion. "Why not?" "You can make seven!" "No you can't!" I turned to Melly and said, "Well, why can't we, Melly? Lots of people won with 8 yesterday." She paused, "There's no seven on the die, Mrs. D," with such assurance, as if I had completely lost whatever sense I had. I shook my head in amazement. She was right. For just a moment I had forgotten the numbers we were working with! I knew if I had made the error, others in the class had made it as well. I wanted everyone to realize that it was an easy mistake to make. I needed everyone to know I had made a mistake, but I wanted it to be an "ah hah!" moment: I said out loud," Of course you make eight by adding one and seven. It's just that when we're using dice, there is no seven on a die. It's easy to forget that using the dice changes how many possibilities we can make." Almost everyone nodded, even the students who were confused at first. "Remember, everybody....what numbers are on the dice?" We counted off the numbers one, two, three, four, five and six on our dice. Light dawned. "We don't have a seven!" I had to say it a couple of times and wrote the numbers on the die on the board: "We can make eight by adding one and seven, but not if we are only using the numbers on our dice."

We finished the lesson by finding all the combinations you could make using the numbers on the dice to make eight, nine, ten, eleven and twelve, and the students noted how the possibilities decreased after seven -- "It looks like a mountain!" they said of the way the combinations of sums built up to seven and then ... as the possibilities decreased..."went down."

To explain why this lesson meant so much to me, I have tried to capture the excitement of the students as they worked through this process. I'm pretty sure some of the students got a glimpse of the beauty of this exercise. From the looks on a couple of faces it seemed they were a little bit shocked to see this process unfold, surprised at how it worked and interested by how cool it was. Other students thought we were practicing addition facts and got into the swing with excitement. Everyone thought it was pretty funny that Mrs. D. made a mistake, but that I did should't come as a surprise to them.. . Everybody in 1D should be starting to get the idea that mistakes help you learn.

So when Ms. B. left that day after we finished our Lab Books, it hit me that it really had been two remarkable math days. For our class to pull apart such a big mathematical idea with so little fuss was quite an accomplishment. We did do some really good math that day.