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.

How do we measure liquids?

Friday I brought a container of water to math group and asked the students how we could measure a liquid. One student suggested that we could measure the temperature with a thermometer (calling up a S'math lesson we did earlier this school year.) Another said we could find out how much it weighed. Then someone pointed out it would be hard to find out because if we weighed it on a scale, we would also be weighing the container it was in. We discussed how to get around that problem. Then someone suggested measuring liquid with a measuring cup. I brought out some measuring cups and liters, and we looked at those.

Then I brought out a tall, thin glass and a short, fat glass. I asked which they thought would hold the most water. One student said he thought they would hold the same amount. The others were divided between thinking the tall glass and the wide glass would hold the most. When we carried out the experiment, it turned out they each held exactly 1 cup.

As we moved on the Math Choice time, some students chose to experiment some more with the water and the different containers.

Measurement Explorations

We have been working on finding items in the classroom that are about as long as the white and orange cuisinaire rods. Once students found 5 items for each, we learned that the white rod is one centimeter long and the orange rod is 10 centimeters or 1 decimeter 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 last Friday, we developed a clear sense of the relationship between an inch and a centimeter.

This activity was drawn from Marilyn Burn's work.

Even and Odd and Crackers

Winter break is a wonderful time for I get to spend lots of time with family. One day at lunch my almost five year old granddaughter, Rachel, was staring intently at her cracker. Then she announced that it had an odd number of holes. "There are seven holes, and seven's an odd number." How do you know, she was asked. Well she had partnered up the holes and there was one left over. She said her teacher had taught her that. So hooray for Rachel's teacher who knew preschoolers are ready to learn about numbers and hooray for Rachel for taking the odd/even lesson and testing it out in real life. She continued her explorations of odd and even over our time together.