Mathematics and Bananas
Dan Meyer notes that mathematics is all around us if we are willing to “notice” it. His interest was piqued when he looked at Ilona Vashchushyn’s Twitter feed that took a random “noticing” in her own classroom of a student eating a green banana and turned it into a survey: what ripeness of banana do you prefer to eat? Maybe amazingly, literally thousands of people responded with the answers corresponding to a nice bell curve.
Just to note that if you are a math teacher and like to ‘follow’, both Meyer and Vashchusyn are worth following.
The banana problem inspires, Vashchusyn suggests, some interesting potential teaching opportunities:
- Have students predict the shape of the distribution.
- Design and carry out a similar survey that they believe would get a similar distribution.
- Connect the process to probability.
- Reverse the task by presenting the findings first and asking what the graph is about.
- Test the data for normality (what a typical distribution looks like)
Interestingly, Alison Gopnik in the amazing The Philosophical Baby shows that babies also deal in statistics and probability (p. 81ff) and from 2.5 years of age can make “genuinely causal inferences”. There are actually some amazing experiments well laid out in that book. It would seem that these ideas are not limited to high school or college students!
But to go back to “noticing”, it is not self-evident that examples from the world around us are necessarily more interesting or engaging than examples from a text-book. Unless the examples from the world around us are connected to the students, they are potentially just as esoteric. But connected to the students themselves, or derived from the students themselves, these kinds of examples can be exciting and totally absorbing. See here for an example from Dan’s own classroom as to how he integrates volume, translation, arithmetic, research, collaboration using a setup that is so intriguing that every student is involved.
My son used to come home from school in 6th and 7th grade and be totally frustrated with his math homework, often opening up the textbook and bursting into tears. He actually knew very little math or, at least, was so emotionally overwhelmed that any math he did know was hidden deep beneath his emotional wreckage. New school, new teacher and a miracle happened. He came home and I asked him what he was doing in mathematics. Algebra, he said. And then he volunteered this information – we are using examples from an old college textbook because the school textbooks are too “childish” (his word).
“Noticing” thus can bring the subject alive for each student, avoid the tedium of endless worksheets, and speak to the student without being patronizing. Vashchusyn is probably right when she comments that taking from the student’s own perceptions and interests is intrinsically more engaging than “noticing” stuff as you walk around in the adult world. Both can, however, be incredibly absorbing if they are quirky enough (for middle schoolers), challenging and controversial enough (for high schoolers), and part of their imaginary world (for elementary age children).
It is interesting too that Meyer and Vashchusyn both blog about their experiences. In other words, they take seriously the need to move to the metacognitive in order to improve their own teaching. In those blogs, they share their experiences, reference other sources, and accept sometimes fierce criticism as being part of the improvement journey. “Noticing” then includes being in the public space so that what you are thinking and doing can be tested in the crucible of your peers’ own experiences and thinking. If you don’t have time to blog yourself, join theirs and be part of the conversation!