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.

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.

Gone Fishin'

In S'math we are doing an activity from AIMS Education Foundation (Activities Integrating Math & Science) that helps show the effects of camouflage. Half the students have done the activity and the other half will do it next week.

Students "fish" for paper fish that are spread out on a construction paper ocean. There are equal numbers of four colors of fish: red, white, blue (the same color as the "ocean"), and black. Teams of students take turns fishing and record their results. Then the whole group meets together to share the class results. We put them on a big chart on the chalkboard so that we can easily compare the number of fish caught. Inevitably either the red or white fish are caught the most. Usually the blue is caught the least, though a few times black has come in last in number caught. I don't tell the students ahead of time that this is a camouflage activity. It prompts some "aha" moments as the students consider why we get the results we do.

I have learned to trust the activity and not get worried if one team's results are skewed because black or blue is a favorite color of some of the fishers. Though I tell the kids not to try to pick up certain colors, a few will do that. Fortunately the rest of the group balances them out. (Or maybe it's the kids who like white and red best!)

World Math Day

Wednesday, March 4, was World Math Day. Although we did not participate in any of the online competitions associated with Math Day, we found a fun way to celebrate the day. Lower School head teacher, Lisa, came to Sky Class and taught the students a math game based on some traditional Native American math games. After playing the game with a partner, students got to take the materials home so that they could play the game some more.

100 Equations

On our 100th day we began a chart to see if the class could come up with 100 equations that equal 100. There was a lot of initial effort and now it has ebbed somewhat. But slowly we are creeping up to 100. As of now we have 98, so just 2 more to go! We've got an interesting combination of addition, subtraction, multiplication, and division equations and some combos. It has been fun watching the kids work out equations. Some just think of one and then look to see if it's on the chart already. Others will take an equation already on the chart and vary it a little.

100th Day!

Monday was our 100th day of school. We had a fun morning doing a variety of activities centered on the number 100. The kids always love making 100 day Trail Mix. They get to choose 10 different items (from a selection of dried fruit, bite-sized cereal and crackers, seeds, & chocolate chips) and get 10 of each item. They line their choices up on a hundreds chart then slip it into a bag to munch on. Another favorite is the 100 pig hunt. While one teacher reads Pigs in Hiding, the others hide 100 numbered paper pigs around the main part of the classroom. Excitement ensues as the children try to find all the pigs. As they are found, their numbers are turned around on a hundreds pocket chart. We are now down to about 8 unfound pigs. Some years the last ones don't turn up until end of year cleanup!

Every student had a collection of 100 items to put on display. They all arranged their collections along with a written description. River Class came to visit, and we had a chance to view our classmates collections. It was a wide variety that included collections of shells, rocks, Hershey's kisses, drawings of flowers, pictures created in Photoshop, legos, coins, etc. Students did 100 foot relays, played games with a hundreds chart, colored 100 hearts and 100 stars, worked to write 100 words, and strung necklaces of 100 beads. We are still working on filling a chart with 100 equations that equal 100.

Pictures: Upper left: making 100 item trail mix Upper right: game with the 100 chart
Lower left: a building for 100 frog counters

"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!

S'marvelous S'math

We include activities in our curriculum that we call S'math because they use science and math together. One source for ideas is AIMS Education Foundation (http://www.aimsedu.org/). AIMS stands for Activities Integrating Math & Science. We also draw on other sources to "cook" up our plans, drawing in any themes we are pursuing as well as student interests.

This year our whole Lower School theme is "Way to Grow." Some of our early S'math activities focused on plants, using our Sky Class garden at times. We are also studying the human body and learning about its systems and parts. One activity focused on the skeleton as we explored bones and joints. More recently we did an activity (drawn from Marilyn Burns) in which students were asked to estimate how many times a strings equal to their heights would go around their heads at the forehead. We then went on to explore other body ratios (How does your height relate to your wingspan? How does your height relate to the length of your leg?)