From Counting to Times Tables

Posted February 11, 2020

In Appendix D of Visible Learning, the authors outline one of the “Eight Effective Mathematics Teaching Practices” – use and connect mathematical representations”. This means that as teachers (parents included), we teach and present the concrete (“I see two dots”) which leads to the understanding of the concept (knowing two dots is “two”) which leads to the procedural knowledge of math (one dot and one more dot is two dots).

10 Frame

Start with the concrete and conceptual. At school, we start counting the days in Kindergarten using 10 Frames. Check out this activity from the NCTM (National Council of Teachers of Mathematics)

Think of the counting you can do! Start with some objects at home and organize them using the ten frame! In this way, students learn that one is always one, 4 is always 4, for example. They learn they can organize items into groups to count them efficiently.

Below is a picture of an activity in grade 3. The student is learning that there are different ways to represent different numbers. This conceptual understanding leads to the procedural  – “ 5 and 1 more is 6” and from here students can be introduced to writing the math fact 6 = 5 + 1.


To encourage students to apply this knowledge, they can be challenged by asking: ”How many ways can you show 7?”

To get from addition to times tables, take the tens frame as 2 groups of 5 = 10. At home, you can make equal groups of objects to teach repeated addition: 5 + 5 + 5 = 15, 3 groups of 5 = 15, 3 x 5 = 15. 

From concrete, to concept, to procedural.


Visible Learning in Mathematics, Hattie, John, Douglas Fischer and Nancy Frey, Corwin Mathematics, 2017.

Tens Frame from:



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