Even and Odd and Crackers

Winter break is a wonderful time for I get to spend lots of time with family. One day at lunch my almost five year old granddaughter, Rachel, was staring intently at her cracker. Then she announced that it had an odd number of holes. "There are seven holes, and seven's an odd number." How do you know, she was asked. Well she had partnered up the holes and there was one left over. She said her teacher had taught her that. So hooray for Rachel's teacher who knew preschoolers are ready to learn about numbers and hooray for Rachel for taking the odd/even lesson and testing it out in real life. She continued her explorations of odd and even over our time together.

Equal Sign Explorations

Last spring when I attended the NCTM conference in D.C. I went to a workshop on Number of the Day for Algebra. The leaders of the workshop shared some research done with elementary math students. Students were given the open number sentence 8 + 4 = ☐ + 5. They were asked what number should go in the box. Not only did a majority of younger elementary students provide a wrong answer (usually answering either 12 or 17), but a majority of students through sixth grade did. Fascinated by the workshop, I sought out the book they referred to in their presentation, Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.

So how can we help students develop a more accurate understanding of what the equal sign means? The book suggests in part a framework of using true and false number sentences and “open number sentences” (such as in the research above) to get children thinking and talking together about this topic. For children who “get” what the sign means and what that means in terms of the relationships between the numbers involved, it provides a chance to solidify their thinking as they work to explain it. For children who do not fully understand what the equal sign means, it provides a chance for them to hear their classmates explanations, which can help their developing understanding.

I have begun exploring true and false number sentences with my math group. Over time I will also introduce open number sentences in which they need to provide a missing number. It will be interesting to see how these activities and the math talk that they generate will affect the student’s thinking about numbers and equality.

Smathing Pumpkins

In Sky we do math in different ways at different times of day. It doesn't just happen in Math Group. It is a big part of what we call "S'math," science and math together. We draw a lot of our activities from AIMS Education Foundation (http://www.aimsedu.org/). AIMS stands for Activities Integrating Math & Science. We also draw on other sources, and we like to draw in any themes we are pursuing as well as student interests.

Last year we did an AIMS activity about apples. This year I thought I'd put a twist on it and do it with mini-pumpkins. After dividing into groups of 3 (at least one boy, at least one girl, at least one old-timer, at least one new-timer) we worked with the mini-pumpkins. First we looked at them with our "scientist's eye," observing closely and noticing details. Each student drew the group's pumpkin, using observation skills. Then they came up with estimates for how many teddy bear counters they thought it would take to balance the pumpkin in a scale. Once estimates were done, each group worked together to weigh the pumpkin and record how many teddy bear counters it took.

The last part of the activity was to think about the distance around the pumpkin, the circumference. Students estimated how many teddy bear counters would make a train as long as a string that went around the pumpkin in its fattest part. They also estimated how many unifix cubes would make a train that long. Then they worked together to cut a string that went around the pumpkin and to do the actual measurements and recording.

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.

Visualizing Numbers

One thing math teachers learn is that it is important for children to be able to represent and visualize numbers in different ways. Last spring when I went to the conference for the National Council of Teachers of Mathematics, I heard a lot about visualizing numbers in a variety of sessions, including one on Singapore math and ones presented by Marilyn Burns and others from Math Solutions. (For more about my thoughts on the NCTM conference see http://skyclassjoanmath.blogspot.com/2009/04/nctm-meeting-in-washington-dc.html and http://skyclassjoanmath.blogspot.com/2009/04/marilyn-burns-and-more.html )

Lisa Rogers from Math Solutions presented a session I enjoyed on developing number sense. One model she shared for visualizing numbers is using ten-frames. It's a model that helps students regroup numbers within our base 10 number system. I have done several mini-lessons with my math group using the overhead projector to introduce them to ten-frames. I have drawn ideas from Susan Scharton's book, Teaching Number Sense- Grade 2. The children have responded with enthusiasm and have been becoming more articulate in describing how they see and manipulate numbers with ten-frames.

Frog Riddles

It's a new year and a new math group. Some of the students were in my language group last year. A couple are new to our class. Others I know as they were in Sky last year, though not in one of my small groups. As I often do, I have chosen an early activity from Marilyn Burn's and Bonnie Tank's A Collection of Math Lessons from Grades 1 through 3. Basing my lesson on the chapter on "Riddles with Color Tiles," I do Frog Riddles. First students use frog counters in four colors to come up with solutions to clues I give about what combination of frog counters are in a bag I have. They work in groups of two (generated by drawing a colored cube from a box). After we do a couple of riddles and talk about the process, each group is asked to come up with their own riddle of at least 4 clues. Each clue should move the solver closer to the solution, and the solver should be able to get the solution by the last clue. It takes some thought, logic, and planning to develop a riddle. After each group completed a riddle, we exchanged bags and clues and tested each others' riddles. Some were returned to the pairs that developed them for additional clues or clearer clues. Each partnership ended up with a successful riddle.

Marilyn Burns and More

One of the highlights of the NCTM Meeting for me was the presentation by Marilyn Burns on the topic of "Using Assessment to Guide Grades K-6 Mathematics Instruction: A Focus on Number and Operations." I have long used books by Marilyn Burns as a resource for my math teaching. She is a wonderful presenter, engaging with a warm sense of humor. Yes, she did use Powerpoint, which she noted was a recent change for her. But it was an outline which she fleshed out in her talk. She also incorporated some engaging video of the kind of assessment she was talking about. I particularly enjoyed seeing her use an assessment that I have used before, taking it from one of her books. Seeing the questions she asked, the way she encouraged the student to verbalize her thinking and reassured the student that the information would help Marlilyn be a better teacher, was quite helpful. All in all an inspriring, productive hour.

Later in the day I attended a session with a colleague of Marilyn Burns at Math Solutions, Lisa Rogers. She was addressing the development of number sense with primary students. By modelling some of the strategies she suggested and drawing in the audience with questions, she kept me alert and involved even though her session was late in the afternoon.
I also enjoyed a presentation on Singapore math, which is creating a buzz right now in this country. A lot of what was presented to us described a central focus on problem solving, the importance of helping students visualize math, and the laying of a foundation with informal experiences before introducing formal treatments. It is also a spiral curriculum that comes back around and revisits and reenforces topics. Other presentations I attended explored the development of algebraic thinking and the development of an understanding of base 10.