Math Journal

In my math group students often use their journals to work through and show math solutions. I base many of the problems we use on  Cognitively Guided Instruction (CGI) . CGI is a process developed by Elizabeth Fennema, Thomas Carpenter, Penelope Peterson, and Megan Franke and outlined in their book Children's Mathematics: Cognitively Guided Instruction.

It works to build on the math knowledge that students have. A part of the process is for students to solve a variety of types of problems using strategies that make sense to them: modeling and manipulatives, counting strategies, and combining numbers in different ways. Students work to communicate these strategies in their journals. We then share strategies together, so students can learn from their classmates’ process. 

I often work to use the students' names in the problems we solve. That adds to their enjoyment of the process. For some the problems are challenging, and they work hard to come up with their solutions and record them. Other times students easily come up with answers in their heads. Then the challenge is to understand the process they used and represent it in their journals.

This Year's Birthday Graph

Yikes! Is it really near the end of October?

Here is this year's class birthday graph. We do this early every school year. First we line up the names of the months along the carpet. Then the students line up behind their birthday months. We make observations such as which months have the greatest number of birthdays and the least. Then each student colors in a birthday cake and glues it above the correct month on a piece of posterboard. Here is the result.

Measurement Explorations

We have been working on finding items in the classroom that are about as long as the white cuisinaire rod. Once students each found 5 items (everything from a rainbow cube to a pet rat’s ear), we learned that the white rod is one centimeter long. Another day we did the same activity using inch cubes.

After these explorations, students had to measure some things at home with rulers or yardsticks for homework. Where possible they were to measure in English and metric units. As we shared the results of their measurements, we developed a clearer sense of the relationship between an inch and a centimeter. We completed charts in our journals where we estimated and then measured parts of our bodies, such as length of hand, length of foot, and circumference of head. We have continued our measurement explorations by beginning to talk about the concept of area and perimeter, applying the idea to items such as our classroom bench and rug.

Exploring Probability with Dreidels

In December we did a probability study with dreidels. We talked about the tops used for the traditional Hanukkah game of dreidel and what we would expect with a fair dreidel. Students said that each of the four sides should have an equal chance of coming up. One student calculated that if you spun a dreidel 100 times, it should land on each letter about 25 times.

In our study we compared plastic and wooden dreidels to see if one kind was more consistently fair than the other. Students spun the dreidels in class and at home for homework. In the hectic times leading up to Winter Break, we did not get all of our data entered into our spreadsheets. In January each student recorded results in spreadsheets on a classroom computer. We then viewed the results in a pie chart and as a bar graph. We speculated about why one side might come up more often than the others. Students thought about how they were made, about whether the letters were painted on or were on the side in relief, and about how the dreidel behaves when it is spun. I shared that to get an accurate picture we needed to have a lot of spins. We ended up with about 700 spins on plastic dreidels and over 900 with wooden dreidels. Here are our results:

Stick-to-itiveness

It is interesting to see how students respond to a problem that they cannot figure out quickly. Some move to “I don’t understand” and get stuck there. Others have more stamina for taking on a challenge. They have a stick-to-itiveness that I want all of my students to develop. I want them to learn how to play around with a problem that is hard. The challenge is how to help them do this. I encourage a lot of experimenting and modeling with manipulatives as one way to get them thinking through problems. Viewing problems as puzzles is another way to get them to relax and explore possibilities. Hearing the approaches of other students can help them broaden the strategies they know how to use. I work hard to create an atmosphere where it is okay to take a risk.

When we were last visiting our granddaughters (and their parents, of course), I watched as the six year old once again untangled the Newton’s Cradle that her sister had once again gotten tangled. She clearly viewed it as a puzzle and patiently observed the way the cords connected as she worked out the tangles. A few times she was on the verge of tears, but she calmed herself and proceeded. Once it was restored, she proclaimed that after a challenge, her brain felt good. My hope is that all students can feel the pleasure of taking on a challenge and not just the frustration, perhaps seeing that the frustration can deepen the pleasure when they find a solution.

My husband’s poetic musings on this are on his poetry blog.

Exploring Shapes

This fall we have been looking at shapes in math group. First we looked at some different triangles and worked to come up with a way to describe a triangle that made it clear what a triangle is. I talked about this as a definition of a triangle. Working in pairs (generated by pulling colored cubes from a box), students shared ideas of what they noticed about the triangles. Then those pairs shared with the others what they came up with. Everyone had noticed the three sides right away. We talked about the difference between an open and closed shape. Several students mentioned the three corners. I introduced the term angle. Then we looked at the word “triangle” and broke it down to tri and angle. One student quickly made the connection to tricycle. That same day we looked at squares and talked a little about how we could describe them.

A few weeks later we revisited the square discussion. This time I drew a square on the board along with some other quadrilaterals. I asked them to work independently this time to describe what a square is in their journals. It was interesting to see how closely some observed the shapes and used terms that sounded more mathematical while others used more visual terms. For example in trying to distinguish a square from a parallelogram with equal sides, several students wrote that a square is not “squished.” Others said the square did not have diagonal lines as a way to describe the same attribute.

The next time we met we shared our ideas. We revisited the idea of angles. At that point one student made the connection to Logo programming which he had heard about from his older brother. We then explored the idea of different sized angles. Some students were recognizing the distinct angle in a square, so I introduced the term “right angle.”

Counting Around the Circle

A new warm-up routine that I am using with my math group comes from a book I read over the summer, Number Sense Routines by Jessica Shumway. The routine is “counting around the circle.” I choose a student to begin the counting and tell them how to count (by two’s, three’s, five’s, ten’s, etc.) At first we just practiced counting in different ways. Now as we get ready to begin counting, I ask them predict what number a certain student will say. Or we count around the circle once and I then ask what number we will get to when we go around a second time. It is a fun way to wake up their number sense.

Frog Riddles

Last week we did an 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.

World of Math

On the first day my new math group met, I asked the students to think of all the things they could that were part of math. As they suggested things, I made a “map” of the World of Math. If we think of more things to add as the year goes on, we will put them on the map as well. I was happy to see that they were able to think of a lot of things to add in addition to arithmetic. Patterns, problem solving, and fun all had their place in the map we created.

Counting

On Thursday Tom took the old-timers and the new second year students to the multi, which gave me some time with our first year students. One of the students had repeatedly asked if we were going to do another scavenger hunt. (There was a Lower School scavenger hunt at our Open House for new students on Monday.) I came up with a number scavenger hunt that involved counting various things in the classroom. As the students went about counting and recording, it was a great chance for me to see how different ones took on the task. I observed how they went about counting the objects and how much confidence they showed in the counting. One student proudly shared that he counted by 3’s. I also watched as they recorded the results. A student asked if 8 was the number with “two balls.” Two girls consulted together about whether the 1 in 12 should go first or not. They had fun with the activity and seemed satisfied as they turned in their recording sheets.

Math Games Online

I have been working on assembling links to some good online math games to put on the computers that the students use at Centers time. We already have KidPix, Logo, and a couple of installed math games on the computers. The links will expand the choices.

As I evaluate which games to use, I found an online article that has been helpful. It was posted by NCTM (National Council of Teachers of Mathematics) and discusses math games and how to evaluate the many online games to finds ones that support students’ learning of math concepts in an interactive way. Here is a link to the article, which includes links to some games that they recommend:  http://www.nctm.org/resources/content.aspx?id=27612

I follow the Free Technology for Teachers blog which features a wealth of resources for teachers available online. Richard Byrne, who writes the blog, periodically includes math resources, including games, in his post.

Turtle Fun

Catching up on activities from November and December:
One focus was learning about a simple computer programming language called LOGO that was developed at MIT as an educational aid for children. In Logo the cursor is a “turtle” who moves across the screen creating graphics based on the commands given by the children. We learned some basic Logo commands, and students then worked on writing out programs (a series of commands) which they could type into the computer to see the results. This project has generated a good deal of excitement in our group. Initially the students worked in partnerships, helping each other as they learned. Then each student had an opportunity to work on a laptop solo. Logo helps students develop their spatial/geometric sense as well as providing good problem solving experience, as they try to figure out how to make the turtle create the drawings they want.

Dreidel Probability

In December we did a probability study with dreidels. We talked about the tops used for the traditional Hanukkah game of dreidel and what we would expect with a fair dreidel. Students said that each of the four sides should have an equal chance of coming up. A few talked about how they want gimel to come up the most. Then we speculated about why one side might come up more than the others. Students thought about how they were made, about whether the letters were painted on or were on the side in relief, and about how the dreidel behaves when it is spun. I shared that to get an accurate picture we needed to have a lot of spins. In our study we compared plastic and wooden dreidels to see if one kind was more consistently fair than the other. Students spun the dreidels in class and at home for homework. Then each student recorded the results in a spreadsheet. Here are our results:

Correcting Math Misconceptions

Our Head Teacher has recently shared some new books with us on the topic of math. Two that have piqued my interest are Math Misconceptions: From Misunderstanding to Deep Understanding by Honi J. Bamberger, Christine Oberdorf, and Karren Schultz-Ferrell and the related book, Activities to Undo Math Misconceptions by Honi J. Bamberger and Karren Schultz-Ferrell.

I have been reading the section on two digit addition and subtraction. The writers share this from research:
 “When children focus on following the steps taught traditionally, they usually pay no attention to the quantities and don’t even consider whether or not their answers make sense.” (Richardson, 1999,100)
and they add that“an understanding of place value is critical to computing efficiently and effectively.”

It gives me good feedback to see that a number of activities they suggest to counter the misconceptions are ones that are part of our curriculum. We include activities such as noticing patterns on a 100s chart, representing two-digit numbers using different sets of tens and ones with cubes and blocks, and having students verbalize and share the strategies they use for two-digit addition. I also have found some new activities to add to our explorations. In one, Make 100, students roll two dice and get that number of Unifix cubes. They place them on a mat with a place for ones and tens. When they get enough cubes, they can snap them together to make a ten and move that group of ten to the other column. They keep going, recording each turn, until they reach 12 turns or 100. For students who need additional challenge, they can work to mentally compute how many more they need to reach 100 at each turn. This game is similar to a game I usually have the students play with base 10 blocks, in which students trade in 10 cubes for a 10 stick. I like the version with Unifix cubes as a lead-in to the other game, because the students are constructing the tens themselves. Later we will play a game that takes away cubes, so that they can deconstruct tens as they subtract.

Math Journals and the 0-99 chart

I use math journals in a lot of ways. Often I ask children to record their math thinking about a group problem solving session in their journals. They also record math explorations and math game results. Recently we have been working to place numbers on the 0-99 chart.

The first day I presented the chart, I had already placed 0-16. I asked how many number cards were placed on the chart. Some students thought 16, but then someone pointed out that 0 was a number, too, so there were 17 cards on the chart. That day we placed 17 more numbers. At our next math group I turned the chart around so that they could not see it. I reminded them that there had been 17 cards on the chart to start with and that we had then added 17 more cards. How many cards were on the chart now? I asked each student to figure out the answer to the question and to show how they figured it out in their journal. Some students found a way to represent the cards in a drawing or by marks and added them up. Some used grouping by tens to help with that. Others worked with equations. Some needed a little help to talk through their process before writing it down. Some came up with the correct answer and some did not, but all were able in the end to represent the math process. The next math group we counted the cards to check our work. Those who came up with an incorrect answer were able to see what tripped them up. I told them that my goal was that everyone be able to show their math process in their journal. Everyone met that goal.

Birthday Graph

Our new school year is well underway. The first week, as we are getting to know each other, we do a graphing activity with the whole class. First we place the names of the months of the year in a row along the rug. Then we call out the months, one at a time, as the students sit behind the name of the month in which they were born. Once our "people graph" is complete, we ask the students questions based on our graph. Which month has the most birthdays? Which has the least? Is there a month with no birthdays? and so forth.

Next each child chooses a birthday cake picture to color in. These are labeled with the children's names and birthdates. Then each is placed above the correct month on a poster board chart. Now our birthday graph poster is prominently displayed in the classroom. The children refer to it on a regular basis to check on birthday's coming up and what cakes each child chose.

My Math Explorations

Today as we settled back into our routines after Spring Break, I took a little time to share with my math group about some math related explorations I did over break. For the first part of the break I took time off entirely: time to play with my granddaughters, time to watch the Final Four, time to hike and look for wildflowers. But the last few days of break I began to look through some math materials and think more about math and the rest of the year.

Math Solutions has a new book that looks interesting, Faster Isn't Smarter. Several chapters of the book are available online (http://www.mathsolutions.com/index.cfm?page=wp18&contentid=994&crid=294&mcrid=107) and that gave me a chance to read through them and mull them over. I shared with my students the name of the book and the idea that sometimes thinking deeper and taking time to process yields a broader understanding than coming up with a quick answer. Being fast in math doesn't necessarily mean you are smarter than others who take more time. I told my students that what is important is to learn what strategies work for you.

I also shared an activity from the latest edition of Teaching Children Mathematics, published by the National Council of Teachers of Mathematics. The activity I read about was for kindergarten students so I upgraded it some for my students. It involved tossing some 2 sided counters (red and white) and recording how many of each color turned up. Students were asked how they figured the totals (recognized the number, counted, knew the number fact, grouped the counters in their mind, etc.) They got practice in explaining their math strategies to the group or to a partner. Some students took on the additional challenge of explaining what they heard another student share as a strategy. We will do this activity again soon, working with a different total number of counters.

More Fractions!

On Monday I got out some play dough and shaped it into squares to represent brownies. Then we worked together to figure out how we could cut each "brownie" into fractions, such as fourths, sixths, and fifths. As students worked, I asked them to put into words the strategies they were using. At the end, the students agreed that it was easier to cut the "brownie" into an even number of pieces, as they could start by cutting it in half. Fifths proved the most difficult.

Then we learned the game Uncover from Marilyn Burn's About Teaching Mathematics: A K-8 Resource. It is the opposite of the game Cover Up which we learned a couple of weeks ago. This time we started with our whole piece covered by the two half pieces. We rolled the fraction cube to see which piece we could remove. Students had a choice on each turn of removing the piece that represented the fraction they rolled, trading one of their pieces in for equivalent pieces (e.g. trading a half in for a fourth and two eighths,) or passing.

Fraction Games

We have several board games in our classroom that focus on fractions: Frog Pond Fractions (by Trend Enterprises, Inc.,) Auntie Pasta's Fraction Game (by Learning Resources,) and Pie in the Sky (by Learning Resources.) Now that we are working on fractions in math group, the games are on our math group shelves and are one of the choices that students may make at "math choice" time, when they have finished their group work for the day.

Fractions

We have been exploring fractions lately. We started out cutting up an apple and talking about what fraction of the apple the pieces were. The students were able to recognize that two parts needed to be equal for them to be halves, and so forth. We also worked with blackberries to begin talking about fractions of a group of things. The additional value of these materials was that we could eat them when we finished.
We went on to make fraction kits, an activity from Marilyn Burn's About Teaching Mathematics: A K-8 Resource. In the activity, the students cut 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.

After we made our fraction kits, we learned the game Cover Up, in which they roll a fraction cube to get pieces to cover the whole piece. On Friday we used the kits to explore equivalent fractions as well as how fractions add together. Later we will learn the game Uncover. It is a great hands on activity.