Saturday, October 27, 2012

Mathematical Thinking

The post below was cross-posted on my classroom blog. It was written for parents but I thought it might provide a good follow-up to my previous post here about math.

Mathematical Thinking
We are learning to share our ways of looking at mathematical situations and ways of solving mathematical problems with each other. We are noticing that most of the time there is more than one way to solve a problem or look at a situation. When we listen to each other explain how we figured something out we learn new ways to think about things. This helps us to be mathematically flexible which helps us not "get stuck" when working on math problems. Below are some photos of one way that we look at a mathematical situation - seeing dots on dot cards - in different ways.

We do a short (5 minute) daily activity that involves looking at dot cards. I flash a dot card picture (as seen below) to the crew for a few seconds. After I cover the picture, I ask the crew what they saw. They then tell me how many dots they saw and how they saw it. We have several students share how they saw the dots. Here are a few examples:

One student saw this as the same configuration of dots on a domino. She knew that this was 8 because this is how 8 appears on a domino.

Another student saw this as a row of 3, a row of 2 and another row of 3. He said that 3 + 3 + 2 = 8.

Another student saw the 6 dots on the sides like they are arranged on a 6 domino. She then added the two dots in the middle to get 8.

The next student saw the 3 + 2 + 3 =8 pattern but she saw it in columns instead of rows.

Another student remembered that we had previously seen a 9 dot arrangement on a previous day and noticed that this was the 9 dot pattern with one dot missing in the middle so it had to be 8.

Why do we do this? I explained to the crew that it is similar to reading - we can "sound out" every word we read but that would not be efficient. We need to have a bank of words that we just know so reading can move along quickly. This is true in math, too. We need to have strategies to deal with numbers so we don't have to count every dot to know how many dots there are in a pattern. We become more efficient using numbers.


This activity also helps us be able to break numbers apart into chunks that are easier to work with in our head. This makes us more flexible when adding and subtracting numbers. We will eventually begin to move on from dot cards and to visualizing numbers and basic addition and subtraction facts. Having the dot patterns in our minds can help us make that transition.

One other reason to do this short, daily activity is that it stimulates our creativity. We begin to look at things in more than one way. This helps us be open to finding more than one solution to our problems.


Thursday, October 11, 2012

Reaching Understandings

Students who struggle in math often lack number sense.

As students build their number sense, mathematics takes on greater meaning. Mathematics becomes more about reaching understandings than following rigid sets of rules. With strong number sense, children become more apt to attempt problems and make sense of mathematics. It is the key to understanding all math.
                                                                      -Jessica Shumway in Number Sense Routines (p.8)

These words tell exactly how I feel about math and math teaching and learning. The phrase "reaching understandings" is just where our focus should be in math. All too often we go straight to "following rigid sets of rules" and this is where we lose students. The "rules" are complicated and meaningless to most students. We need to be helping students explore their own ways of solving problems rather than memorizing sets of rules.

I compare this to reading instruction. What do we really think matters the most in reading instruction? I think most of us would agree that comprehension or making meaning of the text is our ultimate goal. We have learned that focusing only on the rules (phonics, etc) in reading can and usually does lead to readers who tend to be word callers - those who read words correctly but don't understand what they are reading. We know that we need to focus on the meaning of the story being read along with using some decoding strategies. Often the most successful decoding strategies are based on using meaning - thinking about what the story is about, using background knowledge, using context clues, using picture clues, etc. We need to do a blend of meaning and rule following in order to become readers, but we always need the emphasis to be on meaning.

Having number sense and "reaching understanding" in math is similar to comprehension in reading. We may need a blend of rule following and understanding to become mathematicians, but the emphasis should always be on understanding. Always. While not a popular argument, students will be able to use calculators and computers to do those tasks that fall under the "rule following" category, but they will still need to be able to decide what makes sense in a given situation or if the "answer" that they get from their calculator or computer is reasonable. When students have number sense and deep mathematical understandings they will not only be able to decide if an answer is reasonable but they most likely won't need a tool to help them with the computation very often.

I am passionate about this and have been working to make shifts in my math teaching to ensure that students have time to develop deeper mathematical understandings. Thankfully, there are many books out there to help (written by other passionate math teachers). The picture below shows some of the books I have read that have influenced my thinking. I would highly recommend them all. Who or what is influencing your mathematical thinking right now?


Number Talks by Sherry Parrish
Math Exchanges by Kassia Omohundro Wedekind
Number Sense Routines by Jessica Shumway
Teaching Student-Centered Mathematics by John A. Van de Walle

Wednesday, October 10, 2012

Making Connections


I firmly believe this statement that is posted on the wall in my school. It is true that you must develop a deep and positive relationship with each student that you hope to teach. Sometimes, though, those connections can be difficult to make. This year I have two students who I had trouble making a connection with but blogging and Elephant and Piggie came to my rescue. Both students tended to be watchers and not joiners. They both held themselves back or talked about how they didn't like anything that was going on in the classroom. It was obvious that they were scared of what might happen if they let themselves join in an activity. Thankfully two things happened in our room that each of them found simply irresistible. 

One thing that happened was that each student was able to start their own KidBlog. This was the key for one of these students. He loved having a blog and wanted to post all the time. He even posted from home. He loves getting comments and faithfully answers them. One day he even asked if he could stay in for recess and work on his blog. He and I began to have conversations in the comments of his blog and these conversations continued in person each day. Another day, he knew that I had asked him a question in a comment on his blog. He realized that it was not his day to have a turn on the computers in our room so he wrote a book with his answer to my question during our Writing Workshop. I always speak highly of the connections we make through our blogs but in this instance it was not a long distance connection. It was the "way in" for me to be able to reach this student who was right there in the room with me.

The other thing that happened was that we began an informal study of the Elephant and Piggie series by Mo Willems. These books are always very popular with my first graders. This time these two characters gave me a "way in" with a different student. This student was even more reluctant to join in activities than the previously mentioned student. The draw of Elephant and Piggie books was too much for him, though. He fell in love with the books and one day came to whisper in my ear. He quietly asked if he could take one of the Elephant and Piggie books home to share with his parents. The transformation in this student was more subtle than the change in the other student but it was still significant. He began to want to write books during Writer's Workshop and eventually asked if he could put his books in our classroom library for others to read.

Both blogging and Elephant and Piggie books are something that I have always loved and shared with others but after these experiences, I am thankful to them for giving me a "way in" to be able to connect with these students. Now I must go and order the latest Elephant and Piggie book so that I can have it in my classroom when we return from Fall Break.