Author Archives: Beverly Ford

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.

Sparks in the Desert – A Beginning Conversation on Counting On – Part 2

Are there any dangers in training your students in the “strategy” of counting-on? After reading Dr. Les Steffe’s work, I would argue it is harmful. He calls counting-on a non-teachable scheme.  This means that if you want counting-on to be meaningful for students you can present situations that would promote their construction of counting-on, but… Continue Reading

How Cooking Helped me Relate to a Child’s Experience in Math

In my last blog I talked about how the research I have been studying focuses on the “mathematics of children” and I claimed that research that articulates “mathematics of children” can provide powerful tools for a teacher. Many of us experienced elementary school a long time ago and this creates a challenge for our teaching.… Continue Reading

Addressing Mathematical Practice Standards Through Multiplication and Division Word Problems

Addressing Mathematical Practice Standards Through Multiplication and Division Word Problems

Have you ever given your students an experience with manipulatives and then found when you shifted over to a textbook that the students didn’t make the connection between the two experiences? As a curriculum developer and researcher, I am constantly looking for more ways for students to make connections from the concrete (manipulatives) to the… Continue Reading

Building Confidence in Math with Multiplication

Building Confidence in Math with Multiplication

Why do you teach? I remember when I first came into the profession it was because I enjoyed students and wanted to make a difference. I still love watching movies of teachers that have gone into challenging situations and inspired students to think differently. These teachers empowered the students to be all that they were… Continue Reading

Writing a Multiplication Word Problem

Writing a Multiplication Word Problem

Word problems are typically not students’ or teachers’ favorite part of the math lesson. When I talk with teachers, they are frustrated with teaching multiplication word problems. I think one of the things we have been missing is teaching students the structure of what is involved in any multiplication word problem. “Look for and make… Continue Reading

Partitioning Shapes: Is it Geometry or Fractions?

Partitioning Shapes: Is it Geometry or Fractions?

How early should we teach words like half, thirds, and fourths to children? I know that I have often heard that we give young children things they are not developmentally ready for, and I agree. But when it comes to having language identify a concrete experience, I think children can handle it. I was measuring… Continue Reading