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.


  1. Lack of number sense is the biggest challenge that HS math teachers face. I love what you've done. I know that many (most?) of my Geometry and Algebra 2 students didn't have someone who did those activities with them (or they didn't do them enough).

    I looked back on your previous math post and saw the books that you referenced. All listed as elementary material. K-<6.

    What do we do with 13-18 year olds who still don't have that number sense but have made their way to these levels of high school math? I did the SET puzzle of the day today and was thinking about how good that and MasterMind and the like would be for my students. But how do we do that in the midst of our already packed curriculum? And if we DO choose to spend a few minutes here and there on things like this, how do we make them not seem to 'babyish'?

    1. I wish I knew how to make number sense activities not seem "babyish" to older students. I teach first grade now but I taught 8th grade math and algebra for the biggest chunk of my career - at the beginning of my career. I think that is why I am so passionate about doing number sense activities with my first graders. I know how important it becomes as they move through the years.

      The basic idea of a number talk would probably not be to "babyish" for older students. You would just need to adjust the numbers that they work with to more grade level appropriate ones. There would still be many ways to think about or solve a problem and lots to discuss. Not sure if this helps but it is one idea. Thanks for reading.

  2. What a great explanation of why we teach subitizing! The parents of your students are lucky to have such a thoughtful, informed teacher for their students.

  3. Jill, I too have been doing something very similar with my students after reading Number Sense Routines by Jessica Shumway this past summer. Her book is filled with simple ideas just like this that get to heart of important concepts in mathematics. I love how you compared it to sounding out every word in reading and I'll be using that with my students too. Thanks for sharing. Karen

    1. Karen,

      I do love finding ways to make connections between the content areas. It is almost a hobby for me. :) Thanks for reading.