Rachel Counts (in more ways than one!)

I am enjoying Winter Break, a chance to catch my breath in the midst of a busy school year. We are in the Smokies with family for the holidays. This has given me lots of good time with granddaughter Rachel, who reports her age as three and three-quarters. Like her father and uncle when they were this age, she delights in patterns and numbers. We count pennies as we play dreidel. We count blocks and toys and the cookies we cut out of the sugar cookie dough. We count cards as we play Concentration (Memory), where she shows a strong visual memory. Her number sense seems to strengthen every day. We were building with her baby sister's blocks one day. We had established the previous day there were ten of the soft rubbery blocks in all. We were trying to recreate a structure we had built the day before, but could not as some were missing. She carefully counted out eight blocks. I asked her how many were missing. She thought for a moment. "Two! " she announced. Later we found one of the missing blocks and added a bear to the structure.

Who Knows How the Time Goes?

Telling Time is a usual topic for kids this age. Over the years, as digital clocks have become more common and analog clocks have become rarer, I've had to give thought to how much emphasis to put on telling time with an analog clock. While a digital clock is easier to "read," an analog clock, to my mind, gives a clearer sense of the passage of time and fractions of an hour. So while some of my students report that they do not have even one analog clock at home, I continue to spend some time on working with analog clocks. We use mini instructional clocks so that each student has one and can move the hands as we try to both set certain times and read times from the clock. I think one reason learning to read an analog clock is challenging for some students is that the numbers mean different things depending on whether you are looking at the hour hand or the minute hand. For concrete thinking seven and eight year olds, this can take some getting used to.

A Whole Lot of Counting Going On

And measuring and estimating and adding and multiplying and spotting shapes! On Thursday we had Math Day in Lower School. The morning was devoted to a variety of math activities. Each student got to choose 3 activities in which to participate, everything from Math Basketball and Math Baseball to Tangrams to Brain Teasers to Math Games. I offered a Math Scavenger Hunt. Students were challenged to count and measure various things on the playground and in the class. Some of the things they counted generated numbers to add and multiply. There were jars of items for estimation and counting. They looked for where they could find different geometric shapes on the playground and made a face with shapes. Fortunately the threatening clouds held off with their rain until afternoon, so students could measure the alligator sculpture in the Dragon Garden and do the other outside tasks. It was a great way to celebrate Math.

Turtle Explorations

We have been 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 have learned some basic Logo commands, and students are beginning to write out programs (a series of commands) which they can then type into the computer to see the results. This project has generated a great deal of excitement in our group. It has also been our first time to use the new mobile laptop lab which has added to the excitement.

Real life math is not always neat and tidy

Today we did a journal problem that can sometimes trip up some of the students. Here is the problem:
Four friends found a bag with 18 marbles in it on the playground. They told Lisa, the Head Teacher, about it. She told them that if no one claimed the marbles in 2 weeks they could keep them. No one claimed the marbles. How can the friends share the marbles fairly? Use numbers, words, and/or pictures to explain your answer. Be sure you show what happens to all 18 marbles. Remember that in real world math, things do not always work out neatly and evenly.

One of the students called out after I read the problem, "Gee that's hard." One quickly wrote this solution: 18 (divided by) 6 = 3. When I asked him how that related to the numbers in the problem, he said, "Oops!" and went back to work. Things got pretty quiet as all the others settled down to work. Everyone was able to come up with a solution that accounted for all the marbles. Several suggested giving the extra two marbles to Lisa. One suggested giving them to two other friends. A couple needed a nudge to make their solutions clear. A lot of them used diagrams to show clearly their results. On Monday we will share solutions.

Countdown!

We usually begin math group with a whole group activity. After that we often work with partners or individually. Children often finish the work/project for the day at different times. If there is time left before snacktime at 10:00, they may have Math Choice. Choice time usually includes Job Cards (individual math task cards from Creative Publications and Marcy Cook), math workbooks, and math games from our math shelves. Currently the games on the shelf are Race for a Dollar and COUNTDOWN!.

COUNTDOWN! (by Cadaco) is a current favorite. In this game each player has wooden pegs numbered 1-10. In turn, players roll two dice. Each takes the two numbers generated and adds, subtracts, multiplies, or divides the two numbers. He/she turns over one peg representing the result of the operation chosen. For example, if 2 and 6 are rolled, the player can flip over 8 (2+6), 4 (6-2), or 3 (6 divided by 2). Multiplication does not work here as 2 X 6 is more than 10. The goal is to flip over all 10 number pegs. Some of the children are quicker at mental math and easily come up with the options. Others are helped by their fellow players to consider what numbers they can use. Over time they develop strategies, recognizing which numbers are harder to get and turning those over as soon as they can. It is a great game for strengthening students' understanding of number operations and their quick recall of math facts.

Challenge and Initiative

One of the problems in homework that was due on Friday asked the students "How big is your bike?" "Can you solve this problem?" The problem was from the Creative Publication's Primary Math Problems of the Day. There had been some earlier problems that provided enough information for students to solve the problem as well as some that did not. This was the first really open-ended problem in this area. Quite a few of the students found a way to solve this, measuring the length and/or height of their bikes using tools such as a ruler or a tape measure. One student reported that she did not have a bike. A few students had simply written "No" as an answer. When questioned, they said they did have bikes, but either they were not sure what to measure about the bike or were unsure what tool to use. "I know we have a tape measure somewhere, but I didn't know where it was." 

I used this as an opportunity to talk about "Initiative." I told my students that they are all smart (which is true!), but being smart only gets you so far. You also should take initiative and think creatively. I then asked the group to think about tools and materials we have used to measure things before, which include a variety of standard and non-standard measurement tools. I also asked what additional things we could use if we did not have a ruler or tape measure at home. Students came up with ideas using everything from string (you could cut a piece of string as long as your bike, then bring it to school to measure) to a crayon (see how many crayons long your bike is.) I then asked the bike owners who had not found a way to say how big their bikes were to take the sheet home and measure their bikes this weekend.

Riddle Follow-up

On Wednesday and Friday last week, I had the group serve as "testers" for the riddles the partners had written on Monday. (See the post "Math Riddles.") Each worked with his/her partner from Monday. We took a riddle and went through the clues one by one. Each clue narrowed down the possibilities of which frog counters were in the bag. Once we had gone over all of the clues, we determined if we were left with only one possible solution. Two of the riddles succeeded in revealing the solutions. Three needed one or two more clues. Usually the partners who wrote the riddle were able to come up with additional clues that would work. Where help was needed, members of the group suggested ideas. It was a great example of collaboration.

We enjoyed all of the riddles. Each set of partners had set about the task in their own way, yet all found success in the end.

Friday Math Homework Review

On Fridays we take time to review the math homework for the week. For the daily homework, I often draw on Primary Math Problems of the Day from Creative Publications and Marcy Cook. Problems include estimation problems, as well as ones that direct students to make a drawing that shows the solution and to write an equation for the problem.

When we review the problems, I ask students not only to share their answers but to share the strategies they use to get those answers. It helps them learn to put their math thinking into words and lets them see that there are often different approaches/ strategies that can work. For example the estimation problem this week was to estimate how many students in our class have a piano. Several students articulated (each in his/her own way) that they thought of particular students in the class who they knew had a piano and then added some more, thinking that if these students had one that there were probably some others they did not know about. Another simply said he knew some had pianos and some did not, so he just guessed a number that he thought was not too big and not too small. Interestingly, I thought that students who had a piano at home would be more likely to give a higher estimate, but this was not always true.

Math Riddles

Today I did one of my favorite Marilyn Burns activities: Riddles with Color Tiles. I use frog counters which have the same colors as the tiles. We started with a riddle I had prepared, showing them a paper bag in which I had put some frogs. They worked with partners determined by drawing cubes out of a box. I gave out the clues one at a time, having the partners talking together and agreeing on a guess to display. By the end every group had correctly guessed what was in the bag (4 blue and 2 red frogs.) Now came the more challenging part. Each group was to choose some frog counters (10 or less and no more than 2 colors) to put in a bag. Then they had to write a series of clues that would by the end reveal what was in the bag. It was fascinating to watch the partners work. Some needed reminders to include their partners in the work. Others settled right down and cooperated well.

Next time we will test out the riddles to see if they work!

Exploring Geometry

Math learning follows a spiral. Children revisit topics over time, hopefully learning more and deepening understanding with each visit. All of the seven and eight year olds in my math group have had experience with shapes in general and pattern blocks in particular. Last week I pulled out the pattern blocks and had my group examine each piece. An activity like this is an opportunity for me to see where different group members are in terms of their math understanding of a topic. It also allows me to work with children at different levels of understanding as I am covering both basic information as well as some more advanced concepts.

Everyone could identify the name for triangle. Then I asked them what made a triangle a triangle. All knew it had 3 sides. Anything else? Do they know what the word triangle means? How about the prefix tri-? After several guesses, someone made the connection with tricycle. Then we looked at angle. Several had a sense of what an angle is, but could not quite put it into words. Once we got a better sense of angle, and I provided some words for the term, I asked about the "Triangle Area" that we live in. Several were able to identify the points of our geographic triangle (Durham, Raleigh, and Chapel Hill.) We went on and identified, described, and explored the other shapes: square, rhombus, trapezoid, and hexagon. Hexagon was identified by several as either a pentagon or an octagon, so we got to talk some more about prefixes as well as stop signs. Children this age love to play with shapes, so before we went on to do some of Marcy Cook's Young Geometry problem solving tasks, I gave them a chance to make their own shapes. As always the Marcy Cook activity was fun for them, and as I watched them work, I learned more about how each of them approaches such tasks.