Tricks in New York
Several things converged today – in my life as a math teacher, as a reader, as a person who wishes the world was a better place for those who are being left behind by educational and societal systems. The seed for this convergence was a twitter conversation I participated in (or maybe just lurked around) on racism and privilege, in which @sophgermain recommended the book Why Are All the Black Kids Sitting Together in the Cafeteria by Dr. Beverly Daniel Tatum. I purchased the book and have begun reading it, stopping frequently to copy down quotes on post-its and reflect on uncomfortably resonant truths.
Similarly, last night I attended “How I am Working to Learn to Suck Less” at the Global Math Department with @sophgermain. I listened to the presentation with my daughter, trying to think about ways in which I might be contributing to racism, committing micro-aggressions, and engaging in culturally insensitive behavior, loathe as I am to imagine that I am doing any one of these things. I take Soph’s first suggestion deeply to heart – Educate Yourself – and have a long reading list already. (I keep hearing my child Geo telling me “Just google it, Mom,” when I asked for enlightenment on non-binary gender identity.)
Then today, I was sent from my school to a central grading site to participate in the scoring of the open ended questions on the NYS Geometry Regents exam. I was assigned to grade two questions, one of which was the last question on the exam – the big 6-pointer, which was, as it often is, a proof. What was unusual – and mind-boggling to this math teacher – was that the proof was a FILL IN THE BLANK question. That’s right – a 9-step proof on similar triangles in which all of the statements were provided for the students, and three of the nine reasons. I won’t go into my lowering of expectations rant right now, but know that it exists in my mind.
Before we began grading, we had to ‘norm’ as a group; we reviewed the state rubric, the provided student work, and discussed what answers we would additionally accept that might not be included with the materials from the State Ed Department. The final step in the proof involved the equality of the cross products of the proportion of corresponding sides from the similar triangles, and a lengthy debate took place over whether ‘cross multiplication’ was a legitimate reason for that step. I led off the more ‘conservative’ side of the conversation, and pointed out that Cross Multiplication was merely a procedure – and a trick (right out of Nix the Tricks) rather than a bona fide justification for taking a logical step. I was surprised (naively, perhaps) to hear a good portion of the teachers in the room disagree with me, the rationale being that students were taught for so many years that Cross Multiplication was a mathematical ‘idea’ that it might not be reasonable to expect them to be able to state “In a proportion, the product of the means is equal to the product of the extremes.” The debate went on for almost 30 minutes. I am a firm believer in the idea that there is always more than one way to approach a problem in math, and that students should be encouraged to express their mathematical thinking in all of its diversity. But I also have strong feelings (clearly) about what constitutes actual mathematical reasoning, and the net downward effect of lowering expectations so far that real critical thought is no longer required. In short, the conversation left me surprised, and well, shaking my head.
As we began to grade, the long debate became moot in many cases; very few students were able to complete the proof, and even fewer wrote anything which resembled an appropriate reason for that last step. As anyone who has graded these exams knows, you can become downright awed by the breadth of misunderstanding and the chasm between what we think we are teaching and what evidence is actually provided by students in their answers. But right after that ‘awe’ follows the sadness that this huge disparity exists, and what it implies about our classrooms now, and the students’ futures. We think – or I think, rather – that there is real teaching and learning, of some sort, going on in my classes, even if my students aren’t articulating the mathematical brilliance which I am certain I am imparting. I think there is something of value that I am passing on to them, a means to make sense of things, that they can use somehow in the future. But there – in those [sometimes unbelievably and creatively irrelevant] answers on the tests is the hard cold truth – I really am the teacher in the Charlie Brown cartoons.
So where is the convergence? It exists in [my mind, clearly] the space between the need for us to be intentional in our behavior and in our teaching, in the idea that I have to acknowledge and learn about the system of which I am a part, the system which maintains advantage and privilege through oppression, and I somehow have to turn that awareness into meaningful teaching of mathematics for my students. This thought struck me so forcefully today – how critical it is that I try to provide them with some type of tool in the form of math to help them make their way and rise up against the odds. I don’t know how to do that – it’s the question I have been trying to answer for eight years now – but every time I allow myself to fully look around, its exigency hits home again. The future belongs to all of us, and to ignore this pressing need condemns not only our students, but all of us, to a dangerous world.
Her Mathness 
Good ponderings. Unlike you, I teach in rural America. Poverty is a huge issue. Hopelessness is a huge issue. Realizing that if you get an education you’ll have to leave because there are no jobs is a hard truth. Like you, I have no answers as of yet.
Great post straight from the heart. Sadly, there is no easy answer except to keep on the path that you know is the right one…developing math concepts deeply and connecting them to prior learning, without introducing the tricks. My approach is to keep on being passionate about what I do with the understanding that my work will filter through to a few.
I’m on the other end my career. I no longer have my own classroom. I instead work with teachers and try to impart the same ideas to teachers that you are trying to impart to students. The more teachers I can help to see the underlying concepts and connections, the more students will. That is, if the teachers buy in to my “conservative” approach. When you mentioned that many teachers disagreed that cross multiplication was not considered a valid reason in a proof, I was not surprised at all. In my experience, there are many teachers who do not know mathematics deeply. They themselves use the tricks and teach them to the students. They do all the talking and thinking and do not expect students to practice “create viable arguments and critique the reasoning of other” or any of the other CCSSM Practice Standards. It is how they learned and how they are passing on their knowledge.
It is how I learned to teach as well, but my passion for mathematics education has made it important to me to continually keep up with new ideas and change my ways. It is a huge undertaking to change the way that many teachers interpret and impart mathematical knowledge. There are more teachers, young and old, than you realize who have not made the transition to approaching mathematics education in a different way, a transition that has been going on for the last 10 to 15 years. Luckily there are a group of young teachers like you who are coming into the ranks fired up and ready to do the right thing. There are also still a group of us old farts who have seen the light and are trying to influence others. It’s an uphill battle, but one in which I am totally engaged and will continue to be.
Elaine, thank you for reading my blog, and your thoughtful comments. As I begin to plan the spring term, it feels exhausting at times to reimagine my classes again, but also exciting as I employ new ideas.
You flatter me, Elaine, to assume I am a ‘young’ teacher; I am early in my career (this is my 8th year teaching), but I am what is known as a career changer; I’ve had at least 2 other careers, and I am 53 years old.
Your Mathness wrote: “I was assigned to grade two questions, one of which was the last question on the exam – the big 6-pointer, which was, as it often is, a proof. What was unusual – and mind-boggling to this math teacher – was that the proof was a FILL IN THE BLANK question. …
As we began to grade, the long debate became moot in many cases; very few students were able to complete the proof, … ”
Queries: How many proofs of theorems usually appear on a New York State Geometry Regent?
Can a student game the system, by skipping all proofs of theorems, and still pass the Geometry Regents?
(My Geometry course [at Erasmus High School in Brooklyn] was a deductive proof of a theorem a day. But, that was more than a half century ago, so maybe I should not trust my memory.)
How many lesson are allocated to deductive proofs of theorems these days?
My concern is that only allocating a limited time to deductive proofs of theorems may be insufficient for many students to become fluent and comfortable with deductive proofs of theorems.
Jerome – thanks for stopping by. There is usually just one formal proof on the Geometry Regents exam these days, although there is often also a coordinate geometry proof. Many of the questions are ‘proof-based.’ A student can definitely pass the Regents without being able to write a proof, although they need to have a pretty decent knowledge of geometric properties and objects.
I went to high school in NYS more than a few years ago, and I remember doing proofs ALWAYS. But I have looked at the Regents exam I took (they are all available on the State Ed website if you dig around), and there aren’t that many proofs on it.
These days I try to get students to always justify their thinking and work with theorems, postulates and definitions, even if we aren’t writing two-column deductive proofs.