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.