Building Understanding for the Math Operations
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 guide students in developing an understanding of the operations (and fractions) by moving through a series of steps that build meaning. Each vertical progression helps teachers meet students exactly where they are and move their understanding forward.
For example, the progression for division shows how understanding can grow step by step, and the collage below highlights some of the visuals that make these concepts more accessible for students.
If you’re ready to delve deeper into the progressions, here is the link to the library of progression videos.
You might also like these posts on building understanding for multi-digit multiplication and addition and subtraction.
Bring This Approach to Your School!
If you’re an administrator looking for ways to make math instruction more meaningful and engaging, I offer professional development sessions that show teachers how to move beyond worksheets and build real problem-solving skills. Teachers will leave with practical strategies they can use right away—and students will develop deeper understanding and confidence. Interested? Let’s connect!
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.
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 )
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.
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.
In Georgia, students are not required to master the standard algorithm (division) until 6th grade.
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.
It’s showing 0.56 divided by 4.
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.
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.
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!
I love this picture –just looking at it helps me get through the lesson! You are great!
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.