Back in October I saw a paper, Magnitudes Count in Math Education by Tali Leibovich-Raveh and Simon Gregg. Since I learn so much from Simon Gregg’s documentation of his work with students I was eager to devour this paper. It’s been on my mind as I consider the work I bring to the classrooms that I visit. This paper has given me new things to think about.
The discussion in the paper centers around examining counting and continuous magnitudes. Well, I know what counting is (1,2,3, etc) but what is continuous magnitudes? Good question. A straightforward way to express both of these ideas is by asking “how many,” which is counting, then “how much,” which is continuous magnitude. Think of two glasses of water after you’ve been running around on a hot day. You can pick one glass to drink from. Sounds straightforward. But one glass is larger, holds much more water. Now all of a sudden one glass of water versus the other has a different meaning. The larger glass has a larger magnitude of water, so volume of water in the glass becomes the priority concerning the choice of which vessel to choose.
The reason I am thinking about it is this: the authors are keenly interested in understanding and then building on that which children know intuitively, that which is “more intuitive and acquired earlier than the ability to understand numbers.” This is, in fact, something I’ve had sustained curiosity about: what concepts develop naturally in children, and what work can I do to help build on the foundations that are already in place? Also, what ideas do I not want children to internalize?
Once upon a time I was a child. I remember key moments, moments that marked paradigm shifts which impacted my relationship in the world. Here are two of these moments from early elementary school.
During art class I (nervously) asked Mr. Gunther if I could color the clouds in my drawing blue, and leave the sky uncolored. He lowered himself down to my little desk, sweetly but emphatically stated that this was my drawing, so it was my decision. This changed so much for me, I have no words to express it.
Another moment was during a dry 4 part lecture series that my school made all the students endure. We filed into the auditorium, lights dimmed. for a slide show that I remember nothing about other than one thing that shook my world. The lecture series was about the great religions of the world. I walked into those talks thinking my own sect of my own religion was the biggest religion in the universe, and, certainly, the only true religion. I walked out, shocked to know that I was wrong about this. I suppose I would have eventually figured this out, but knowing about this diversity in the world from a young age made a real difference to me.
These memories are enough to convince me that any amount of time with a child can make a difference to them. I want to be thoughtful and deliberate in what I bring to children.
I’m often on the lookout for paradigms that are infused into us from an early age that may not be broad enough to create a foundation for wider, wiser growth. We learn numbers, we learn to count. Then comes adding, subtracting, multiplying and dividing. I dare say that this is about the place where most adults stopped retaining anything they learned in math class. In fact, this idea that numbers, counting, and arithmetic fully defines math is what I believed for many years. Where I have the opportunity, I want to let children know that math is something more than math facts.
The authors of Magnitudes Count have done three great favors for me.
- They’ve identified a mathematical concept that children innately understand, which they believe can be used as scaffolding to teach more complex concepts.
- While they state that more study needs to be done to investigate the relationship between an understanding of continuous magnitudes and deeper math understanding, the fact that they are pointing us in the direction of emphasizing mathematical thinking besides counting and arithmetic is exciting to me. I didn’t know that people were on the lookout for building on children’s intuitive understanding of math as a place to start
- The authors have provided me with a way for me to introduce my next post, which will be about my own observations about what children intuitively know, innate knowledge that I think about as being useful to use as scaffolding to teach more complex concepts. I won’t have the research or references to back up my thoughts, but I still want to get my thinking out here.
The one thing I would have liked to see in this paper (and it may be in there) is that when students are taught something that is mathematical I think it needs both to be identified as mathematical, as well as to be explained why it is mathematical. By doing this, students may, before too long, understand that math, along with their math facts, can also be about seeing relationships, making comparisons, developing reasoning and all sorts of other good stuff.
One last note, the way I communicate with students is with hands-on materials, mostly paper that is cut, folded, and arranged. It may sound like all I am talking about is abstract, but it all gets translated into projects.
(this is part 2, Here’s Part 1)