Math Talk

I often give my math homework in a "problem of the day" format. I draw a lot of the problems from Think About It! Primary Math Problems of the Day by Marcy Cook, published by Creative Publications. There are different kinds of problems. Some involve estimation. Last week students were asked to estimate how many of our math binders it would take to make a stack 1 meter high. On Fridays we review the homework as a group, sharing answers and strategies. It's a great time for the "math talk" that both helps individual students learn to put math thinking into words and gives students a chance to hear other students' ideas to broaden their perspectives.

The math binder estimation problem sparked a lot of discussion. First I asked, as I often do, what information can help us make a smart guess? Answers included how long a meter is and how thick a math binder is. Then we pulled out a meter stick and tackled the problem of how to find the actual amount. As we stacked the binders we could see that even if we got all the other math groups' binders, we wouldn't have enough binders to get to a meter. Several students showed us their ideas of what we could do. One pointed out that five binders reached the 10 centimeter mark. He explained that there are 100 centimeters in a meter, so 10 groups of 5 would be 50. Another pointed out that 10 binders were 17 cm. and she said we could add by 17's until we got to 100 (though we got to 102 rather than 100 even.) We used both ways and discovered we came up with 2 different answers. What was going on? Several students then recognized that the binders were thicker on one side, so how we stacked them effected the outcome. It was a lively math investigation.

The other problem from this week that sparked a lot of interest was the one that asked how many legs were on 4 octopi. Students were asked to write an equation to represent the problem and answer. Several students volunteered to write their equations on the board, and we talked about the different equations. Two people added up 4 eights, but one got 31 and the other got 32, so we all checked the math. Several had used multiplication equations, and we agreed that 4 X 8= 32 and 8 X 4= 32 were both correct and could both represent the problem. One person had 16 + 16= 32 and a classmate suggested an additional equation of 8 + 8= 16 would make the connection between that equation and the problem clearer.

"But it doesn't come out even!"

This is another journal problem that can puzzle the kids because it doesn't come out even. Today I asked my students to consider this problem:
You invite 10 friends over for a party. You want to serve juice. You want enough for each person to have 1 cup. The extra special juice you want to buy (raspberry razzle-dazzle kiwi!) only comes in quarts. (At this point I asked the students to remember how many cups are in a quart and one of the students supplied 4.) How many quarts do you need to buy? Prove your answer.

Quite a few asked for help. For each I asked leading questions until they came up with an answer and a way to express their understanding of it. Some included themselves in the planning and some did not. One kept trying to adjust the number of people so that every cup was accounted for: "well my parents could come, or maybe I'll have one go out and one stay." That extra cup tends to bother them a lot and I don't think it is in the name of "waste not want not." For them division should be neat and tidy. We'll share our solutions soon. I hope problems like this help them reach some peace with the idea that math can be messy!

Real life math is not always neat and tidy

Today we did a journal problem that can sometimes trip up some of the students. Here is the problem:
Four friends found a bag with 18 marbles in it on the playground. They told Lisa, the Head Teacher, about it. She told them that if no one claimed the marbles in 2 weeks they could keep them. No one claimed the marbles. How can the friends share the marbles fairly? Use numbers, words, and/or pictures to explain your answer. Be sure you show what happens to all 18 marbles. Remember that in real world math, things do not always work out neatly and evenly.

One of the students called out after I read the problem, "Gee that's hard." One quickly wrote this solution: 18 (divided by) 6 = 3. When I asked him how that related to the numbers in the problem, he said, "Oops!" and went back to work. Things got pretty quiet as all the others settled down to work. Everyone was able to come up with a solution that accounted for all the marbles. Several suggested giving the extra two marbles to Lisa. One suggested giving them to two other friends. A couple needed a nudge to make their solutions clear. A lot of them used diagrams to show clearly their results. On Monday we will share solutions.

Challenge and Initiative

One of the problems in homework that was due on Friday asked the students "How big is your bike?" "Can you solve this problem?" The problem was from the Creative Publication's Primary Math Problems of the Day. There had been some earlier problems that provided enough information for students to solve the problem as well as some that did not. This was the first really open-ended problem in this area. Quite a few of the students found a way to solve this, measuring the length and/or height of their bikes using tools such as a ruler or a tape measure. One student reported that she did not have a bike. A few students had simply written "No" as an answer. When questioned, they said they did have bikes, but either they were not sure what to measure about the bike or were unsure what tool to use. "I know we have a tape measure somewhere, but I didn't know where it was." 

I used this as an opportunity to talk about "Initiative." I told my students that they are all smart (which is true!), but being smart only gets you so far. You also should take initiative and think creatively. I then asked the group to think about tools and materials we have used to measure things before, which include a variety of standard and non-standard measurement tools. I also asked what additional things we could use if we did not have a ruler or tape measure at home. Students came up with ideas using everything from string (you could cut a piece of string as long as your bike, then bring it to school to measure) to a crayon (see how many crayons long your bike is.) I then asked the bike owners who had not found a way to say how big their bikes were to take the sheet home and measure their bikes this weekend.

Friday Math Homework Review

On Fridays we take time to review the math homework for the week. For the daily homework, I often draw on Primary Math Problems of the Day from Creative Publications and Marcy Cook. Problems include estimation problems, as well as ones that direct students to make a drawing that shows the solution and to write an equation for the problem.

When we review the problems, I ask students not only to share their answers but to share the strategies they use to get those answers. It helps them learn to put their math thinking into words and lets them see that there are often different approaches/ strategies that can work. For example the estimation problem this week was to estimate how many students in our class have a piano. Several students articulated (each in his/her own way) that they thought of particular students in the class who they knew had a piano and then added some more, thinking that if these students had one that there were probably some others they did not know about. Another simply said he knew some had pianos and some did not, so he just guessed a number that he thought was not too big and not too small. Interestingly, I thought that students who had a piano at home would be more likely to give a higher estimate, but this was not always true.