Greenwich Country Day School, Connecticut, USA
Eric the Sheep waiting in a long line to be shorn becomes Sneaky Pete queuing for ice cream and suddenly we are challenged to consider pedagogical questions.
Our thanks to Marshall who has, in a way, put himself on the line in offering his decisions for collegiate scrutiny. His enthusiasm for teaching and learning is infectious. We would be delighted to receive any views, opinions, comments, reactions from your discussions which could be added to this article.
This article is intended as a teaching craft talking point for your next year level, faculty or staff meeting.
(In the BBC Horizon documentary Fermat's Last Theorem, mathematician Andrew Wiles describes working mathematically as like wandering around in a dark mansion of many rooms.)
- School mathematics is about learning to work like a mathematician.
- A mathematician's work begins with an interesting problem.
- A problem is a problem because it has no known solution and no known pathway to solution.
- Therefore a large part of a mathematician's time is spent working in the dark.
Teaching mathematics in this context involves finding problems appropriate for our students, choosing teaching craft which encourages interest in them and, because classroom time constraints are real, making judgements about how much guidance to give and when. Marshall Spooner is one teacher who has made such judgements and who is happy for you to review them.
Hi, my name is Marshall Spooner, and I am a teacher in Greenwich, CT, USA at Greenwich Country Day School. I am so excited that all of your amazing tasks are now available online and downloadable as PDFs. This is awesome! Please tell me the easiest way to order the eTask library.
I have been at the school for 16 years, but just before I got here, a few of my colleagues worked with you. We still have the bags of some (but not all) of the tasks, and I love the idea that we can now have everything and be more organized! My favorite task is Eric the Sheep which we have affectionately renamed "Sneaky Pete" (see attached) which we use every year with our 8th graders (14 years old), and they love it!
Click an image to view and print the document in full size.
Compare, contrast and discuss these two presentations of the same problem.
Remembering that both presentations of the problem are intended for pairs using concrete materials (to encourage interest), some starter questions might be:
- What might be the pedagogical underpinning of each presentation?
- Too many words?
- Not enough words?
- Why change the context of the task?
- What part do the images play?
- How do kids learn to ask questions if we ask all the questions for them?
- Is a simple task card with an additional Investigation Guide a suitable alternative?
Note: Marshall explains the reasons for his pedagogical choices below. It will be more fruitful to have discussed the section above with your colleagues before reading further.
|I have actually put a fair amount of thought into that problem specifically. I'm attaching my first draft of it, and it's only a subtle change but the numbers that I chose here turned out to be extremely unhelpful examples for the students because they were all multiples of each other. Students would figure out 10 and then 20 and just assume the same pattern applied for 50, 100, 1000, etc.
By choosing some different numbers (and a few more small examples), I find that the students have to dig into the theory of what's happening a little more before attempting to extrapolate to bigger numbers. I guess I may be acting "a little too helpful" in this situation, but when, year after year, the students kept making the same mistake again and again and again, I finally decided to tweak the numbers and it has worked out much better. Also, in 8 years of doing this problem with students, I think only a couple have figured out the very last question, which asks them to generalize any scenarios. It's really quite tricky and a nice challenge.
I hope this helps generate some good discussion. Thanks again for everything!
What do you think?
This space reserved for a report on your discussions of this teaching craft talking point. Email firstname.lastname@example.org.