During our last #ReggioPLC twitter chat I posted a picture of a ball of snakes I stumbled upon while visiting KOLTS Forest School.
Excited by the discovery I called a few children over to see. The conversation took a quick mathematical turn, followed by several theories (and a wonderful demonstration) as to why a snake sticks its tongue out so much.
I approached two children, closest to the snakes and asked if they would like to see them. I asked the girls to walk softly and speak quietly so that we didn’t disturb the snakes in their home. After watching the snakes in silence for several moments one of the girls asked how many there were and I responded, “How do you think we could find out?” The answer not only warmed my heart, it reminded me of my love of the beautiful world of mathematics.
“Well”, responded Anna (names have been changed), “let’s say its a family, and in the family there is a Nona, a Mommy, a Daddy, and a baby. How many would that be?”
“Hmm”, I responded. “Tell me who is in the snake family again?” Anna repeated, “a Nona, a Mommy, a Daddy, and a baby” and as she said the name of each family member, I raised a finger on my right hand to correspond. I waited a moment and then raised my eyebrows at Anna and was greeted by a similar eyebrow raise, followed by the query, “so how many are in the snake family?”
At this point I would likely have used my other hand in the problem solving dialogue but my left hand was being held quite firmly by Lily (name changed). Lily is the strongest and most brave person I know in the forest, however her bravery comes with the need to hold my hand firmly for moral support. Minus one hand, I pondered how to respond. There was no need. Lily cleared her throat and said, “Anna, I will show you how this works. Now, you say the names of the snake family again and Louise you do the thing with your fingers.” As Anna repeated the names of the four family members, I raised a finger for each on my right hand. Without letting go of my left hand, Lily used her free left hand to gently touch each of my fingers as she repeated, “Nona, Mommy, Daddy, baby” and then said to me “keep your fingers up”, looked at Anna and said, “now watch how this works.” As she once again touched each finger she recited slowly, “1, 2, 3, 4. There are 4 snakes in the family.” Anna shook her head in agreement. We watched for a few moments, one of the snakes rapidly stuck its tongue in and out and from there the conversation took a turn to ideas about this snake behaviour.
Later, as I reflected on the snake family moment, I was touched by the central role of relationship in the mathematical dialogue, the assumption by Anna that a group of snakes must be a family and the connections she made to her own family. The relationship between myself and the two girls, from an emotional stance was lovely, but also the challenge we collectively took on as we attempted to utilize “finger math” without the availability of all our fingers, fingers busy with the task of holding hands, a choice that contributed to our well-being and our sense of belonging, cornerstones of our early years curriculum, How Does Learning Happen.
Math itself, particularly numeracy, is about relationships. An understanding of number and quantity for example is developed when we put “all kinds of things into all kinds of relationships.” Remembering this quote, I headed to my bookshelf to grab a few of my favourite math resources. The first was Number in Preschool & Kindergarten (1982) by Constance Kammii. At the beginning of the chapter entitled, Principles of Teaching, Kammii reminds us that although she speaks of teaching number, “number is not directly teachable.”She goes on to suggest six principles of “number teaching.”
- The creation of all kinds of relationships – Encourage the child to be alert and to put all kinds of objects, events and actions into all kinds of relationships. (Anna put the ball of snakes into the relationship of family with the names of four family members).
- The quantification of objects – Encourage the child to think about number and quantities of objects when these are meaningful to her. (Lilly helped Anna to apply a number to each member to determine quantity).
- Social interaction with peers and teachers – Encourage the child to interact with her peers, figure out how the child might be thinking and respond according to what might be going on in her head. (My pause invited Lilly to intervene, both Lilly and I went with Anna’s family analogy to help her with number and quantity). (p. 27)
When we discussed the power of math outdoors during our twitter chat, Nancy made an insightful comment about “teaching” math, in response to Scott who suggested that math outdoors is “experienced” rather than “taught”.
Having read Nancy’s comment, I continued to reflect on math and the way I have most commonly seen it taught. There always seems to me to be a disconnection from children’s innate and intuitive nature to create relationships between things in their world and to express their understanding of these relationships with a more structured, universal approach to “teaching” number and quantity. I knew I would find a pleasing quote from David Hawkins on my book shelf and was rewarded in The Roots of Literacy.
Children’s curiosity and investigative talents can lead them into genuinely mathematical subject matter. This induction can and should take place along with, or even well ahead of, their mastery of the arbitrary, shorthand, written code and rules of operation that we now impose and wrongly call mathematics. There is by now a considerable body of research that shows that a major source of many children’s difficulty in acquiring these arithmetical skills is a matter of unmotivated rote learning. This learning is often dissociated from their native understanding and so also from their talents for extending it (p. 150).
And here lies one of our biggest challenges in the “teaching” and understanding of mathematics. Does it really serve any purpose if we only acquire the ability to perform on a test at some later point in our academic journey? How well are we preparing students to utilize mathematics as an everyday language and an indispensable tool? I thought about these concerns and wondered about the challenges involved in teaching mathematics in a way that each student experiences her own unique cognitive structure. Are we able, as educators to honour and facilitate this necessary freedom?
The object for “teaching” number is the child’s construction of the mental structure of number. Since this structure cannot be taught directly, the teacher must focus on encouraging the child to think actively and autonomously in all kinds of situations. A child who thinks actively in her own way about all kinds of objects and events, including quantities, will inevitably construct number. The task of the teacher is to encourage the child’s thinking in her own way, which is very difficult, because most of us were trained to get children to produce “right answers” (Kamii, p. 26).
I thought once again of Anna and Lily and the family of snakes. I believe in a few short moments, number and quantity was grappled with and grasped.
And so another two weeks have past and Nancy and I are still wondering about the word “teach” in juxtaposition to “experience” in terms of math.
What does it mean to experience math?
What is the difference between experiencing and teaching math?
What is the role of the environment? What is the role of the educator?
How do provocations and invitations invite numeracy? Mathematical thinking?
How might documentation inform our practice in “teaching” mathematics?
Please join Nancy and myself for our next #ReggioPLC twitter chat at 9pm EST Tuesday May 10th to share your thoughts!