Paul Lockhart’s article “A Mathematician’s Lament” speaks of a society where music is treated the way mathematics is currently treated within our school system. He pleads his case for math to be treated as an art form within society, but specifically within schools. While his plea feels too ambitious within the expectations of society (the 4th grade standardized tests, SATs, Compass, MCATs, LSATs, really any other kind of standardized tests), it is nonetheless a culture of mathematics to which I aspire.
“Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.”
I feel very fortunate to teach at Essex Street Academy, one of the members of The New York Performance Standards Consortium, where students’ understanding is assessed through a PBAT (project based assessment task) in lieu of the math Regents exam as a high school graduation requirement. The consortium math rubric consists of the following standards/performance indicators: a. problem solving, b. reasoning and proof, c. communication, d. connections, and e. representation.
In past semesters, problem solving and communication were the two strands I focused on the most for students to become mathematicians* in my classroom.
To explicitly task students with problem solving, I was inspired by Jo Boaler’s aptly named week of inspirational math, and used a similar growing shapes activities by another math teacher from a Consortium school to invite students to, in the words of Paul Lockhart, “cobble together their own explanations and proofs” through the lens of efficient problem solving.
To practice verbal communication of a math topic, my students then participated in student-centered conversations that we call “seminar discussions” at ESA, where the hard work of collaborative sense making of a mathematical text is placed on the students. The clip below showcases their (nervous to be videotaped) collaborative effort at processing which of the problem solving strategies utilized were the most and least efficient (with a link to transcript here if the sound quality is as terrible as I fear it is).
Your PoP feels particularly challenging from a special education perspective – if students are generating their own questions and methods how can you ensure that every student is learning? I know, because I work with your students, that you have developed a tool kit of strategies with them that are widely applicable to many different problems. This has really supported the students with disabilities that I work with closely.
However, I think the missing link for them is the opportunity to practice the same strategy or procedure repeatedly with similar but different questions. For example I would like them to explore in depth: How does creating an equation help us efficiently answer a question about compound interest on a savings account? What do the y-intercept and constant ratio represent in this case? How does creating an equation help us efficiently answer a question about compound interest on a credit card account? What do the y-intercept and constant ratio represent in this case? And so on.
Many students with disabilities need repeated practice in order to master a procedure and to understand why it makes sense. One novel situation is just that – novel, they miss the opportunity to make the generalizations that make these problems interesting and exciting.
I automatically got really excited about this idea, because I love offering students choice. On this big of a scale, I could understand that it seems daunting. Were you thinking of having them work collaboratively with other students who have chosen a problem that will utilize similar skills and content? It feels a little like an independent study in some ways, which would look quite different in the classroom than a “normal” day in math class. How often would they work on these individualized problems and how would you need to change the structure of class / curriculum to allow for this time and shift?
I can’t wait to hear more about how it’s going! I’ve wanted to allow for more student choice within the development of my French curricula, but I haven’t seen a model of it done in a way that allows me to maintain 90% target language and feel engaging in a meaningful enough way.