# Mathematics of Grace: Figurative Mental Material

In my first blog about the Mathematics of Grace, I mentioned that by the end of our six week study she was able to answer 98 + 5. This was exciting for me because when we first interviewed her she wasn’t able to combine 19 + 3. She was limited to solving sums within 20. This was my first experience working with a homogeneous group of students within one of the stages of development from Dr. Steffe’s work. Dr. Steffe found some generalized behaviors in students (the mathematics of students) that helped my team make informed, precise decisions based on what we saw in Grace and what similar students did.

Today I want to write about the difference in the figurative mental material Grace brought to bear in the two methods and show you a video of the most advanced way she responded to an additive situation. If it has been a while or you learn by observing, you may want to go back to my 1st blog and 2nd blog to watch the video of Grace (linked above).

In her first attempt in December of 2015, Grace brought to bear an image of a pattern. She would pop up a finger pattern, then count, and recognize how many fingers she saw. So for 8 + 5, she would lift five fingers on her right hand and her index, middle and ring finger on her left hand saying, “8.” Then she would count and lift a finger one at a time saying, “1, 2, 3, 4, 5.” She would look down at her fingers and answer thirteen. The pattern she could imagine is eight and thirteen. What she could not do is continue counting from eight, five more fingers.

The second time we worked on this, she brought to bear an imagined unit item and could join two imagined collections in her mind to see a new collection that she wanted to count. She only needed to move her fingers and say the number word sequence to have meaning for the 1st addend. She monitored her counting for the second addend with a finger pattern, so she would know to stop counting. For example if she were to answer 27 + 6, she would lift her fingers one at a time saying, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ,11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27.” Then she would “erase” her finger by pulling them into fists and lift one finger at a time to fill the finger pattern for six and saying, “28, 29, 30, 31, 32, 33.” This new mental material and method allowed her to answer what she considered some hard math problems.

In the video above, you can see the delight Grace and her partner Max experience when they respond to what they consider some challenging math problems. You will notice she counts-on to answer 62 + 4 and 98 + 5. I would love to say with confidence that her counting-on behavior could be evidence of her mental material consisting of an abstract unit item, but I had given her the suggestion to not count the first addend. When a teacher or another child is required for the child to have the idea, I would not say the child brings to bear the mental material. I would say they are constructing the mental material.
My next few blog posts will talk about supporting students’ conceptual foundation so that they can construct the counting-on procedure.