Building Understanding for the Math Operations

Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

How did you learn arithmetic? By that I mean basic multi-digit computation, such as subtracting with regrouping, multi-digit multiplication, or “long” division. I learned totally by memorizing a series of steps for each operation. The teacher demonstrated the steps, and we practiced. I’m at good memorizing, so I was able to successfully compute. I’ll be honest in saying that I had very little understanding of what I was actually doing. It was not until I became a teacher and had to explain the process to my students that I realized the complexities of computing with numbers and the need to better understand the math operations, and I am constantly looking for better ways to teach for understanding. Enter Graham Fletcher.

Graham has produced an amazing series of “progression” videos that illustrate how teachers can develop an understanding of the operations (and fractions) by moving through a progression of steps that build understanding. Each vertical progression allows you to meet your students exactly where they are. This week, I am working on division with my 4th graders (whole numbers) and my 5th graders (decimals). I literally studied Graham’s progression for division last week and took screenshots of his illustrations! Here’s a collage showing some of this work.

If you’re ready to delve deeper into the progressions, here is the link to the library of progression videos.

P.S. Wondering how I found Graham and this amazing resource? On Twitter, of course! If you are not using Twitter for personal professional development, you are really missing out!

12 Comments

  1. Susan Jones

    I would love to see some research with adults to see how many of them are missing these concepts — *and* what they remember about learning arithmetic.

    Reply
  2. Carmel

    Hi there
    Love your ideas and suggestions that make maths learning about conceptual understanding and not process. Thankyou .
    We here in Victoria work at differentiating the curriculum based on individual needs and following a pathway of learning. We don’t like to plan for our grade 3 and our grade 4 but rather see the class as a cohort of students with a diverse range of needs and we plan to cater for everyone in our lessons .
    We follow curriculum of course however often students have gaps in their mathematical understanding and unless they are ‘ filled’ it is very difficult to move forward .
    Hope this adds to the discussion in a positive way .
    Cheers
    Carmel ( Maths Coach )

    Reply
  3. Amanda

    Hello,
    I was just wondering if your district requires 4th graders to master standard algorithm division. After watching Graham’s video he doesn’t even show standard algorithm division until 5th grade and I just found that very interesting. I have asked our 5th grade teachers for feedback on what they would like us to ensure the kids come to them knowing and over and over again they emphasize that if the kids master the standard algorithm in fourth grade it makes their job easier in 5th.

    Reply
    • Donna Boucher

      You really have to look at whatever standards your state follows, because they are all slightly different. I would say, however, that it’s not a district decision, but a statewide one.

      Reply
    • Terri

      In Georgia, students are not required to master the standard algorithm (division) until 6th grade.

      Reply
  4. Artie Brunson

    I don’t understand the picture. I see the 4 groups of the rods and cubes. But I don’t see the 0.56

    Explain this to me.

    Reply
    • Donna Boucher

      It’s showing 0.56 divided by 4.

      Reply
    • Lisa

      They are using base-10 blocks to represent the decimal. A flat = 1, a rod = 0.1, and a cube = 0.01. There are 4 rods and 16 cubes total, which equals 0.56. You have to exchange one of the rods for 10 cubes to be able to divide it equally into 4 groups.

      Reply
      • Lisa

        Can you explain how the flat is 1? I have seen the base ten for blocks with decimals in different ways. I am confused on which is correct? I have seen the cube shown as the 1 whole, flats as the tenths, rods as hundredths, and units as thousandths.

        Reply
        • Donna Boucher

          Great question! Actually, any of the pieces can be used for the whole. If the flat is 1, then the rod is a tenth and the unit is a hundredth. If the cube is 1, then the flat is a tenth, the rod is a hundredth, and the unit is a thousandth. You can even make the rod 1 and then the unit is a tenth. It’s all about the relationship between the pieces!

          Reply
  5. Keli

    I love this picture –just looking at it helps me get through the lesson! You are great!

    Reply
  6. Heather Crowson

    I attended some of Graham’s professional development on the foundations of fractions. Great PD! I highly recommend tapping into his knowledge of number sense and fractions. One of the best PD I have attended online.

    Reply

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