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.

NCTM Meeting in Washington, DC

Pictures: Entrance to the NCTM Meeting Exhibits, Convention Center displays

I've recently returned from the National Council of Teachers of Mathematics Annual Meeting in Washington, D.C. I had thought I might be able to blog from the conference but ran into a couple of problems. The first was the limited free internet connectivity available to me, and the cost of the paid connectivity that was offered. As a result, blogging during the sessions did not work. The second problem was I had little time between sessions, as I tried to pack in as much as I could.

So here's my post today. It was a great conference. I had several sessions that were exciting, informative, and stimulating. Several more were interesting and gave me food for thought. A few were rather dull. When will people realize that simply showing a Powerpoint presentation (without pictures or video) and reading it to us with a little extra explanation does not make for an interesting session?

Fortunately most presentations had more zip, including an engaging delivery, thoughtfully posed questions, and for some, audience participation. Another post will detail some of the better sessions that I attended.

More Fractions

One fraction activity that I particularly love with this age is one I take from Marilyn Burn's About Teaching Mathematics: A K-8 Resource. In the activity, the students make fraction kits, cutting identically sized and multi-colored strips of paper up different ways. First they cut one in half and label each piece 1/2. The next strip is cut in 4 equal pieces and labeled appropriately. By the time we get to the fifth piece, the children have cut and written a lot. They are worried that they will need to cut this piece into 32 pieces, so they are relieved that this piece stays intact to represent 1 or 1/1.

Last week we made our fraction kits and learned the game Cover Up, in which they roll a fraction cube to get pieces to cover the whole piece. Later this week we will learn the game Uncover. We will also use the kits to explore how fractions add together and equivalent fractions. It is a great hands on activity.

Fractions

This week I pulled out several board games we have that use fractions. As we looked at them, we discussed what fractions actually are. Students often say they do not know what fractions are. Yet if you ask them to give you half of something they can do it. The gap is between the abstract idea of fractions and their experiential understanding of the world. So part of what we work on is bridging that gap, using manipulatives and talking about what they represent.

Today we used frog counters in two colors to represent fractional parts of groups of objects. If you have 6 frogs and half of them are red, what does that look like? Students made groupings that fit the fraction given and worked on writing the fraction. Can you show what it looks like if one third of them are a given color? Is there another fraction name that would represent the same amount? Some quickly came to 2/6, while others got help from a partner to figure it out.