Author Archives: Beverly Ford
Today I want to write about how the mathematics of students helps me to see my role as a teacher and learning a bit differently. In my last few posts, I have been telling you the story of Grace. Today, I wanted to share Grace’s story because it is an example of a time where I experienced the value of the mathematics of students for myself. Working with Grace was foundational in my understanding of how knowing the mathematics of students could empower me as a teacher and engage with my students in mathematical situations differently than I ever had before.
In my professional learning as a teacher, I was always encouraged to build on what children knew, yet doing this in mathematics meant that concepts had to be taught in a certain order. I knew that students needed to understand addition before they could understand multiplication, but now I see how this applies to what each student knows and has constructed.
This idea of building on what a child knows was much more tangible in language arts. In language arts, my students and I would make a KWL chart about a given topic before we would begin our study (A KWL chart is a table where you list what you know, what you want to know, and what was learned). Understanding how children come to know now allows me to see how powerful and purposeful that experience might have been for my students. For example, if we were studying oceans, I would write down everything that my students knew about oceans. It might range from the animals that live in the ocean to the feeling they had while playing in it. This was their experience of the ocean, and our conversation brought those experiences to the forefront of their minds. We would then ask questions about the ocean and write down what we wanted to know. Sometimes these questions and the resulting answers would be different than what the students expected. We call this process perturbation.
Von Glasersfeld describes learning as what happens when we experience perturbation and make an accommodation. People think of perturbation in a negative connotation, similar to being annoyed, but coming from a radical constructivist perspective, it doesn’t have that same connotation. Instead, from this perspective, it is referring to the realization that what you thought would make sense doesn’t. This is what happened to Grace when her method of answering addition problems wouldn’t work anymore. I intentionally used an example from the research to present her with a situation that would cause her to be perturbed and potentially allow her to reflect and accommodate her addition scheme. You can read and hear more about this through my previous blog post.
If I were to say to a room full of teachers that they need to perturb their students, they might think I am mean or at least odd. When in reality, allowing students to construct for themselves meaningful concepts is the best thing for them. When they make accommodations to their knowledge, they are standing on their own feet, not relying on someone else’s explanation and support to function in their academic lives. I challenge each of you to broaden your definition of perturbation and look for those moments of reflection that happen right after perturbation. Learning has just happened.
One of my favorite questions to ask a toddler is, “How old are you?” They will often proudly hold up two, three, or four fingers. Most of the time these fingers come up all at once. This is their first experience connecting a number word and their fingers and can be a foundation to building… Continue Reading
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.… Continue Reading
In my last blog I wrote about one of the first things I noticed about the mathematics of Grace. She used her fingers to solve addition situations like 7+4 by constructing more advanced finger patterns, where one finger could mean one or eleven and six fingers could mean six or sixteen. This allowed her to… Continue Reading
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… Continue Reading
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
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
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
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
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