A few of us are reading Mathematical Mindsets, by Jo Boaler, as part of a book group on LINCS that will start this coming Monday, April 17th. The book is similar to Boaler’s other writing in that it cites evidence of recent work in brain science to show that everyone can learn, that the brain is plastic and grows like a muscle when used, and there is no such thing as math people and non-math people. Boaler also argues for an approach to developing growth mindsets that is rooted in mathematics. The book includes examples of low-entry, high ceiling problems that can be used to develop mathematical mindsets.
We started with a brief conversation about Cuisenaire rods. Solange has used them to teach fractions and first encountered them in an ESOL class. The rods were invented by Georges Cuisenaire, popularized by Caleb Gattegno, and have been used since the 50’s. Fortunately, BMCC has multiple sets we were able to experiment with.
Then we looked at the Cuisenaire train rod problem:
We first brainstormed a few questions we were interested to answer:
- As we go up different Cuisenaire lengths, is there a pattern in how many ways you can make a train?
- What is the most efficient way to arrange the “cars” in a “train”? We defined the train as the full length (3 in the example above) and car as the rods (that may be different lengths) that make up the train. For example, the green train above has 1 car and the white train has 3 cars.
- What if the shape were 2-dimensional: 2×2, 3×3, 4×4, etc? What if the shape were 3-dimensional: 2x2x2, 3x3x3, 4x4x4, etc? This assumes that we are still building the shapes out of Cuisenaire rods.
We then spent about 45 minutes working on our own. Most people worked on the first two questions. A couple people explored the 2D and 3D questions. While people were working, I individually shared some push and support questions that I wrote, depending where people were in their investigations. I cut out questions as I needed them. (Later in the meeting, we revised some of these questions.)
Solange explains to Kevin her solution for this question: How many trains of length 10 can you make with Cuisenaire rods without using length 1 rods?
Andrew works on the 2D question.
We came together for the end of the meeting and explored Gregory’s organization of the possible trains of length 5. He described it as an alphabetical or dictionary approach. He organized the possible trains by the number of cars in the train. There is 1 train of length 5 (the yellow rod). Then there are 4 trains made of 2 cars. The first car can be 1, 2, 3 or 4. The second car in each train will be whatever is needed to add up to 5. The third row shows the 6 trains made of 3 separate cars. Ruben came up to use Gregory’s method to show the 4 possible trains made of 4 cars. Finally, there is only 1 train that can be made of 5 cars (white, white, white, white, white).
Then Solange and Kevin explained their solution to how many trains are possible of length 10 if you don’t use the white pieces. The three check marks above identify the 3 trains of length 5 that don’t have white pieces. I’ll leave out their solution in case you want to find it for yourself. It’s really surprising!
Finally, Kevin showed us a connection to Pascal’s triangle in the train combinations. I’ll leave that for you to find as well. Check the links for classroom resources and articles about the train problem.
Location: BMCC, 25 Broadway
Attended: Andrew, Eric, Gregory, Kevin, Linda, Ruben, Solange
Programs represented: BMCC, CUNY LINCT, CUNY Start, LAGCC, Hostos Community College