# Author Archives: Beverly Ford

### 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 not before they are showing signs they are ready to make sense of it.

One way a teacher could support students to make sense of problems and persevere in solving them (SMP1) for tasks in which they could count-on is to attend to the mathematics of students and look for evidence that they are ready to construct counting-on. In the unit construction team’s last blog, my colleague Elin began our conversation around this presentation from the recent Palm Springs California Mathematics Council conference. Today I will write about the video we used of Chloe to set the stage for the presentation. Next week you will hear from Grace about the videos we used to highlight some of the progression that leads to meaningful counting-on in students.

Chloe was a 2nd grade student who struggled to solve problems when she didn’t have enough fingers to use in counting. This was an indicator for us that she needed visible things to count, which is the first stage of a counting scheme. Children do not meaningfully count-on until the 3rd stage of development. So, when we noticed her counting-on, we hypothesized that it was something she memorized. We looked for evidence to confirm or deny our hypothesis. In the following video you will see Chloe explain her counting-on behavior. Watch and see what she does.

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Based on what she did in this video, it would be logical to think she was possibly in the 3rd stage of development. As I stated earlier, we had some experience and were skeptical. It is important when gathering data about the mathematics of students to look for multiple pieces of evidence to confirm a hypothesis. Let me explain how she solved another problem prior to this one.

The task: Adding 6 hidden marbles and 3 visible marbles

Chloe didn’t think she could solve the problem. It took prompting by the teacher before Chloe attempted the problem.

Chloe simultaneously lifts a finger pattern for 6 and says, “1, 2, 3” while lifting three fingers sequentially.

Teacher: “How did you know it was 9?”

Chloe: “Because I had nine fingers.”

Teacher: “9 what?”

Chloe: “9…93?”

Now watch this next video clip of Chloe continuing the interview.

Looking at all of the evidence we collected, my colleagues and I argue/infer that counting-on at this point was an empty procedure for Chloe. Chloe’s empty procedure made it possible for her to “solve” problems without making sense of a rather complex situation. She was just following a road map. This circumvented her meaningful development of number.

Stay tuned for Grace’s explanation of the other videos we used at the conference of exemplars on a meaningful journey for a child to count-on*.*

### 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

### The Power of Knowing the “Mathematics of Children”

I have had the privilege of being a part of the AIMS Center from its birth. Having been a teacher, coach, and staff developer for the last 15 years, the world of a teacher is something I have experience with. I am now called a Research Associate and my role is to be a bridge… Continue Reading

### 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

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

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?

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

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

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

### Do We Really Understand What Math Is?

What would you or your students say math is? Some common answers could be numbers, addition, subtraction . . . Below are the posters a group of AIMS trainers created answering that question. Most people don’t understand what math really is. If you have read some of my previous posts, you know my elementary and… Continue Reading