How do you transition from playing math to more formal math activities/lessons?
This question is a companion to my recent Play Power post on the Moebius Noodles blog which provides further context for my question.
In Zoltan Dienes' six-stage theory of learning mathematics the first three stages are free play, learning to play by the rules and the comparison stage. I specialize in this part of the math learning continuum. My approach as a dance educator at the intersection of math and dance/art making is to allow students time to explore the materials and learn basic skills w/in the medium first; as we go along I add the math ideas into the mix a little at a time. By the end of the process we have lovely objects (dance steps, weavings, ornaments, etc) that meaningfully reflect artistic and mathematical ideas.
But, I’ve been curious what comes next and how you make a transition from literally playing with math ideas to more formal math learning. I know that play is useful and important but, honestly, I think it's underutilized, undervalued, and misunderstood, often seen simply as “fun” activities to do as a “break” from the real math. My questions:
If you have used math play in any form (not just art) in your math teaching how have you helped your students make the transition to more formal math activity/lessons?
What can/does this transition between play and abstraction look like in students' thinking and in a classroom/learning context?
Does this progression look different between age groups (young children, upper elementary, middle school, etc.)? How?
The blog post: http://www.moebiusnoodles.com/2014/02/play-power/
The two-year-old playing for an hour:
Here are a couple excerpts from the book Mathematics Their Way, first published in 1976. Although not specifically addressing the question of transition from free play to more formal thinking, I found these to be really interesting and helpful pieces of thinking about free play/exploration. And, it seems that the book itself is a highly specific example of the transition itself over time, from ages 5 to 8. We start with this:
Also: "Some children need more time than others to explore a particular material, so there is no simple formula defining the length of the free exploration stage in each classroom. Try to let the needs of the children determine how long you continue. Children learn easily and joyfully from real things. This free exploration period is the foundation for later development, providing a reservoir of images and recorded sensations in the child's mind; and the only way to build this foundation is by allowing each child to have the time and freedom to explore. One day, or two, is not enough to permit these intuitive understandings to develop and be recorded in the child's mind, and unless they are recorded there, they cannot be drawn on later."
I like @MariaDroujkova's answer. I'm not sure what you mean @dendari.
There are many ways this happens. Game/play/creation in one of the student's main area of interest and they'll pose the questions themselves. There are some games I love because they incite this so naturally. Nim almost always gets the students asking the question immediately: is there a best strategy?
Sometimes you can just ask the questions and they'll start to dig in. Accessibility makes this more likely. Things like finding all the pentominoes.
And sometimes you have to model. It feels like many students have lost their ability to wonder, though really I suspect they've just learned it has no place in school mathematics. I did one of these for the PIG dice game this week. They didn't know math could answer a question like that.
I found it helps to remind students when formal activities are similar to play activites.
Here are the stages we recently developed for Natural Math:
The free stance
0 - Open free play, like a baby
1 - Inspired play, emerging patterns
The maker stance
2 - Noticing, formulating, using patterns
3 - Tweaking and remixing patterns
The mastery
4 - Free creation from scratch, pattern drafting, tool-building
The actions listed under the maker stance, such as formulating and remixing patterns, are situated abstractions, because these patterns apply to each particular context. But at the mastery level, you reach proper abstraction by working with patterns across contexts.
Young kids have some amazing abilities for making bridges across contexts (divergent thinking), but they are weak on systems thinking, which is the domain of grown-ups. Mixed-age groups such as child-parent pairs can produce amazing results at the mastery level!
