Concretely, Pictorially, Symbolically

I had the greatest time seeing Dr. Nicki this summer! She really is a wealth of knowledge, and I got a bajillion great ideas and resources from her.  If you are unfamiliar with her, here is a link to her blog.  She has written lots of books about teaching math, and she just really is ALL THAT.

Anyway, one of the big ideas she left us with at this conference was the fact that students need to learn their math (we focused on facts) in three stages: concretely, pictorially, and symbolically. We often spend so much time on the symbolically part that we forget about the other two stages!  However, these stages are key for ALL grade levels! All kids won't stay in the same stage for the same amount of time, but it is our duty to be sure that they know how to do it and what it means.

For example 2 + 4:
CONCRETELY:

PICTORIALLY:

SYMBOLICALLY/NUMERICALLY:

2 + 4 = 6

I know that I was guilty of glossing over or skipping some of these steps sometimes--esp. with my higher math kiddos. However, research shows that students need to be able to understand each of these steps to gain a deeper understanding of what is happening,  Some students may need to stay in the concrete stage longer than others, but all of our students should be able to explain what is happening in the concrete and pictorial stages.  They may move more quickly to the symbolic stage, but they need to be able to explain how it all works,  Students who just have math fact recall are like students who are good decoders in reading.  They can say what you want to hear, but they have no comprehension of what is going on...

Thankfully EM builds these stages in regularly for our students.  We are so lucky to have such a great resource.  

Dr. Nikki shared a myriad of tools to help us build the concrete part into our lessons.  One thing she had us make was a beaded number line.  It would be a good thing for you to put into your first weeks of school.  Many grade levels would be able to make it independently.  If you don't think  your kiddos could, you could ask for parent volunteers to make them or to help the students make them.  This could be used for all grade levels. Two colors of pony beads and a strong piece of string for each child is all you need. :)

This video has some good ideas for you on ways to use it.

Using a Beaded Number Line from Karen Richardson on Vimeo.


Here also is a virtual beaded number line that you could use along with the kids' own numberlines.  It has some great features, too.

http://mathsframe.co.uk/en/resources/resource/69/itp_beadstring

I'd love to hear how you use these number lines in your room, or let me know if I can help you with ways to put the beaded number line into your lessons.