Author Archives: Beverly Ford

Mathematics of Grace: Using finger patterns

The mathematics of students is a powerful tool for a teacher. It allows a teacher to hypothesize what is happening in the mind of a child and plan a next step that will allow that child to construct more sophisticated understanding. Today I want to look at the mathematics of a student we call Grace (pseudonym). I love telling Grace’s story because she moved from solving problems like 8+4 to 98+4 in our 6 week study!

When Grace was interviewed in December, we noticed that she could solve problems even when counters were not in front of her. She used her finger patterns to help her solve these problems. In my experience working with teachers before joining the AIMS Center, I would have described Grace’s behavior simply as she needed to use her fingers. I had no idea that how a child uses their fingers can be an indicator of very different understandings and mental material. Children use their fingers in a variety of ways to solve problems, and it is helpful when you can use the unique way they use their fingers to interpret their mathematical understandings. .

Please watch the following video and write down a few things that you notice.

Grace did not explain her process right away by using her fingers, but after a prompting from one of the research associates she explained her solution by using a simultaneous finger pattern (lifts all fingers at the same time) and then counting additional fingers. Let’s look at how she solved the first problem, which was seven under one cloth and four under a second cloth. She starts by lifting a finger pattern for seven simultaneously. Then she counts while lifting a finger every time she says a number, “1, 2, 3, 4.” She explains that the solution has to be eleven because she didn’t have enough fingers and had to imagine one more.

Grace is using her recognized finger pattern for seven to help her create something to count. She knows what ‘seven’ looks like on her fingers without having to count them one at a time. In the second problem she solves the problem in the same manner and even states that she solved it like the last problem. Did you notice that detail the first time you watched the video? Watch it again. Can you solve 8+5 using Grace’s method? We refer to mental images, movements, or words as “figurative material”. It is something the student generates (not concrete) to make it possible to solve the problem. Given that fact that she did this without her fingers being visible (they were under the table), I think Grace is definitely moving away from working with concrete material to working with figurative material. Students have to work with figurative material before they can construct abstract material. And constructing abstract material is the goal.

In my next few blog posts I will write about the limitation of this method that Grace used and I will show what tasks I presented her to foster opportunities to construct more sophisticated methods. In the meantime if you’d like to hear more about the mathematics of students, head over and listen to an interview I did with Dr. Chris Brownell and our ZPC Podcast.

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

Three Great Multiplication Posts

Three Great Multiplication Posts

How to Equip Your Students to Better Understand Multiplication, Part One As I have coached and taught in the classroom, the three most popular ways to describe multiplication is showing ______ groups of ______, using repeated addition and making arrays. Now all of these methods have their place in a student’s understanding of multiplication, but… Continue Reading

Finding Math in Unexpected Places

Finding Math in Unexpected Places

I was reading Inchworm and a Half with my 6-year-old daughter, Bethany, last night for the 40th time. She loves reading the section, “Squirmy, wormy, hoppity-hoop! We measure everything, loopity loop.” Even before she could read books she memorized this section and would “read” it. The book is about an inchworm that loves to measure… Continue Reading