In this meeting, we deconstructed a problem from NCTM’s Mathematics Teacher: Learning & Teaching PK-12, using a few ideas from The Art of Problem Posing, by Stephen I. Brown and Marion I. Walter.
Here’s the original problem:
Welcome
Some responses to a warm-up question: Why might it be useful learners of mathematics to pose their own problems?
Katherine: Students must understand the mathematical concept to pose their own problem.
Carol: It’s empowering, for one thing. It’s also rich and challenging form of “thinking backwards” when you construct a problem.
Sharon: It creates ownership- makes it more fun for students.
Gustavo: I think it’s very important. I have my students create homework questions for themselves. Helps me to see what kinds of numbers they select and how comfortable they are in the topic.
Cristina: I agree that writing your own problems demonstrates a deeper understanding of the concepts than only solving problems.
Problem Posing
Launch
Notes from our conversation, including what we noticed and some initial questions (without changing the situation):
Some questions we might pose with most mathematical situations:
We listed some of the attributes of this mathematical situation, then started posing What-If-Not questions by changing the attributes of the situation.
Small groups
Here is some of the work we did in breakout rooms. Group 1 used Mathigon Polypad to visualize the initial situation:
- What does (1 + 3)/(5 + 7) look like with shapes?
- How can we visualize each successive term using L shapes and squares?
- Why does a 1:3 ratio look like a 1/4 in this visualization?
Group 2 changed the attributes of the initial situation in a few different ways:
- What if the series of numbers was 1, 4, 7, 10, … with difference of 3 instead of 2? For example, the 2nd term would be (1 + 4)/(7 + 10).
- What if the series of consecutive odd numbers started with 5 instead of 1? For example, the 2nd term would be (5 + 7)/(9 + 11).
- What if the series of consecutive odd numbers started with 3 instead of 1? For example, the 2nd term would be (3 + 5)/(7 + 9).
Lynda also shared a way of finding the sum of a series of numbers:
Using the numerator of the following fraction as an example…
- Add the outside numbers 5 + 13 to get 18
- Add the next set of two numbers, 7 + 11. to get 18.
- 9 is left on its own.
- 18 + 18 + 9 is the same as 5 x 9. There are five 9’s, which is the number of terms in the expression.
Using the denominator of the fraction above as an example…
- Add the outside numbers 15 + 23 to get 38
- Add the next set of two numbers, 17 + 21. to get 38.
- 19 is left on its own.
- 38 + 38 + 9 is the same as 5 x 19. There are five 19’s, which is the number of terms in the expression.
So, we can choose the middle number in the series of consecutive numbers and multiple by the number of terms to get the sum.
Jeniah shared YouTube videos explaining formulas for calculating these sums:
https://www.youtube.com/watch?v=Z95DnK6KYbI
https://www.youtube.com/watch?v=HHhBg4Cj6jc
