When people ask me what calculus is, my usual answer is “the mathematics of change,” studying things that change continually. Algebra is essentially static, studying things frozen in time and space. Calculus studies things that move, shapes that bend, etc. Algebra deals with things that are exact and consequently can be fragile. Calculus deals with approximation and is consequently more robust.

I’m happier with the paragraph above if you replace “calculus” with “analysis.” Analysis certainly seeks to understand and model things that change continually, but calculus per se is the mechanism of analysis.

I used to think it oddly formal for people to say “differential and integral calculus.” Is there any other kind? Well yes, yes there is, though I didn’t know that at the time. A calculus is a system of rules for computing things. Differential and integral calculus is a system of rules for calculating derivatives and integrals. Lately I’ve thought more about other calculi more than differential calculus: propositional calculus, lambda calculus, calculus of inductive constructions, etc.

In my first career I taught (differential and integral) calculus and was frustrated with students who would learn how to calculate derivatives but never understood what a derivative was or what it was good for. In some sense since though, they got to the core of what a calculus is. It would be better if they knew what they were calculating and how to apply it, but they still learn something valuable by knowing how to carry out the manipulations. A computer science major, for example, who gets through (differential) calculus knowing how to calculate derivatives without knowing what they are is in a good position to understand lambda calculus later.

I’ve been tooling around with the slightly infamous BISG methodologylately. It’s a simple concept which takes the last name of a person, as well as the zip code of their residence, and imputes the probabilities of that person being of various races and ethnicities using the Bayes updating rule.

The methodology is implemented with the most recent U.S. census data and critically relies on the fact that segregation is widespread in this country, especially among whites and blacks, and that Asian and Hispanic last names are relatively well-defined. It’s not a perfect methodology, of course, and it breaks down in the cases that people marry people of other races, or there are names in common between races, and especially when they live in diverse neighborhoods.

The BISG methodology came up recently in this article(hat tip Don Goldberg) about the man who invented it and the politics surrounding it. Specifically, it was recently used by the CFPB to infer disparate impact in auto lending, and the Republicans who side with auto lending lobbyists called it “junk science.” I blogged about this hereand, even earlier, here.

Their complaints, I believe, center around the fact that the methodology, being based on the entire U.S. population, isn’t entirely accurate when it comes to auto lending, or for that matter when it comes to mortgages, which was the CFPB’s “ground truth” testing arena.

And that’s because minorities basically have less wealth, due to a bunch of historical racist reasons, but the upshot is that this methodology assumes a random sampling of the U.S. population but what we actually see in auto financing isn’t random.

Which begs the question, why don’t we update the probabilities with the known distribution of auto lending? That’s the thing about Bayes Law, we can absolutely do that. And once we did that, the Republican’s complaint would disappear. Please, someone tell me what I’m misunderstanding.

Between you and me, I think the real gripe is something along the lines of the so-called voter fraud problem, which is not really a problem statistically but since examples can be found of mistakes, we might imagine they’re widespread. In this case, the “mistake” is a white person being offered restitution for racist auto lending practices, which happens, and is a strange problem to have, but needs to be compared to not offering restitution to a lot of people who actually deserve it.

Anyhoo, I’m planning to add the below code to github, but I recently purchased a new laptop and I haven’t added a public key yet, so I’ll get to it soon. To be clear, the below code isn’t perfect, and it only uses zip code whereas a more precise implementation would use addresses. I’m supplying this because I didn’t find it online in python, only in STATA or something crazy expensive like that. Even so, I stole their munged census data, which you can too, from this github page.

Also, I can’t seem to get the python spacing to work in WordPress, so this is really pretty terrible, but python users will be able to figure it out until I can get it post on github.

%matplotlib inline

import numpy
import matplotlib
from pandas import *
import pylab
pylab.rcParams[‘figure.figsize’] = 16, 12

#Clean your last names and zip codes.

def get_last_name(fullname):
parts_list = fullname.split(‘ ‘)
while parts_list[-1] in [”, ‘ ‘,’ ‘,’Jr’, ‘III’, ‘II’, ‘Sr’]:
parts_list = parts_list[:-1]
if len(parts_list)==0:
return “”
else:
return parts_list[-1].upper().replace(“‘”, “”)

def clean_zip(fullzip):
if len(str(fullzip))<5:
return 0
else:
try:
return int(str(fullzip)[:5])
except:
return 0

Test = read_csv(“file.csv”)
Test[‘Name’] = Test[‘name’].map(lambda x: get_last_name(x))
Test[‘Zip’] = Test[‘zip’].map(lambda x: clean_zip(x))

#Add zip code probabilities. Note these are probability of living in a specific zip code given that you have a given race. They are extremely small numbers.

F = read_stata(“zip_over18_race_dec10.dta”)
print “read in zip data”

names =[‘NH_White_alone’,’NH_Black_alone’, ‘NH_API_alone’, ‘NH_AIAN_alone’,       ‘NH_Mult_Total’, \
‘Hispanic_Total’,’NH_Other_alone’]

trans = dict(zip(names, [‘White’, ‘Black’, ‘API’, ‘AIAN’, ‘Mult’, ‘Hisp’, ‘Other’]))
totals_by_race = [float(F[r].sum()) for r in names]
sum_dict = dict(zip(names, totals_by_race))

#I’ll use the generic_vector down below when I don’t have better name information

generic_vector = numpy.array(totals_by_race)/numpy.array(totals_by_race).sum()

for r in names:
F[‘pct of total %s’ %(trans[r])] = F[r]/sum_dict[r]

print “ready to add zip probabilities”

def get_zip_probs(zip):
G = F[F[‘ZCTA5’]==str(zip)][[‘pct of total White’,’pct of total Black’, ‘pct of total API’, \
‘pct of total AIAN’, ‘pct of total Mult’, ‘pct of total Hisp’, \
‘pct of total Other’]]
if len(G.values)>0:
return numpy.array(G.values[0])
else:
print “no data for zip = “, zip
return numpy.array([1.0]*7)

Test[‘Prob of zip given race’] = Test[‘Zip’].map(lambda x: get_zip_probs(x))

#Next, compute the probability of each race given a specific name.

Names = read_csv(“app_c.csv”)

print “read in name data”

def clean_probs(p):
try:
return float(p)
except:
return 0.0

for cat in [‘pctwhite’, ‘pctblack’, ‘pctapi’, ‘pctaian’, ‘pct2prace’, ‘pcthispanic’]:
Names[cat] = Names[cat].map(lambda x: clean_probs(x)/100.0)

Names[‘pctother’] = Names.apply(lambda row: max (0, 1 – float(row[‘pctwhite’]) – \
float(row[‘pctblack’]) – float(row[‘pctapi’]) – \
float(row[‘pctaian’]) – float(row[‘pct2prace’]) – \
float(row[‘pcthispanic’])), axis = 1)

print “ready to add name probabilities”

def get_name_probs(name):
G = Names[Names[‘name’]==name][[‘pctwhite’, ‘pctblack’, ‘pctapi’, ‘pctaian’,  ‘pct2prace’, ‘pcthispanic’, ‘pctother’]]
if len(G.values)>0:
return numpy.array(G.values[0])
else:
return generic_vector

Test[‘Prob of race given name’] = Test[‘Name’].map(lambda x: get_name_probs(x))

#Finally, use the Bayesian updating formula to compute overall probabilities of each race.

Test[‘Prod’] = Test[‘Prob of zip given race’]*Test[‘Prob of race given name’]
Test[‘Dot’] = Test[‘Prod’].map(lambda x: x.sum())
Test[‘Final Probs’] = Test[‘Prod’]/Test[‘Dot’]

Test[‘White Prob’] = Test[‘Final Probs’].map(lambda x: x[0])
Test[‘Black Prob’] = Test[‘Final Probs’].map(lambda x: x[1])
Test[‘API Prob’] = Test[‘Final Probs’].map(lambda x: x[2])
Test[‘AIAN Prob’] = Test[‘Final Probs’].map(lambda x: x[3])
Test[‘Mult Prob’] = Test[‘Final Probs’].map(lambda x: x[4])
Test[‘Hisp Prob’] = Test[‘Final Probs’].map(lambda x: x[5])
Test[‘Other Prob’] = Test[‘Final Probs’].map(lambda x: x[6])

School district borders often divide students by income — and in Detroit and many other places across the U.S., that gulf is especially wide.

That is the conclusion reached in a report released Tuesday by EdBuild, a nonprofit dedicated to overhauling the way states fund education. The report looked at neighboring school systems and found that the poverty rate can be eight times higher from one district to the next.

“You’re talking about, really, haves and have-nots that are living across an imaginary border that has become very important and has become impermeable,” said Rebecca Sibilia, founder and CEO of EdBuild.

With education budgets funded largely by property taxes, poorer school districts can’t pull in as much money as their better-off neighbors — even taking into account federal aid for poor students. Meanwhile, students who live in poverty often need more resources to succeed in school.

The starkest dividing line in the country separates Detroit from neighboring Grosse Pointe Schools, according to the report. In Detroit, 49 percent of children live in poverty, while the poverty rate in Grosse Pointe is only 7 percent. (Other recent measures of child poverty put Detroit's rate even higher.)

The disparity is deeply rooted. In the 1970s, black parents and the Detroit NAACP sued over racial segregation in the city’s schools. In Milliken v. Bradley, the U.S. Supreme Court ruled that neighboring school districts could not be forced to participate in integration plans. Today, the income disparity between Detroit and Grosse Pointe is even greater than at the time of the court decision, according to EdBuild’s report.

“It is sadly ironic that the number one border remains the border that was decided in Milliken,” Sibilia said.

Separating students by income can have damaging effects in the classroom, said Halley Potter, a fellow at The Century Foundation, a think tank that focuses on inequality, among other issues.

“When you have concentrated poverty you tend to see the weakest outcomes for low-income students,” she said. “Students are missing out on some of those chances to learn from different exposures, different experiences than they’ve had.”

The neighboring school districts with the widest disparity in poverty rates:

• Michigan: Detroit City School District (49.2 percent) and Grosse Pointe Public Schools (6.5 percent)
• Alabama: Birmingham City School District (48.5 percent) and Vestavia Hills City School District (6.2 percent)
• Alabama: Birmingham City School District (48.5 percent) and Mountain Brook City School District (7 percent)
• Pennsylvania: Clairton City School District (48 percent) and West Jefferson Hills School District (7 percent)
• Ohio: Dayton City School District (47.2 percent) and Beavercreek City School District (6.58 percent)
• Arizona: Balsz Elementary District (51 percent) and Scottsdale Unified District (11 percent)
• Ohio: Dayton City School District (47.2 percent) and Oakwood City School District (6.96 percent)
• Ohio: Youngstown City School District (46 percent) and Poland Local School District (7 percent)
• Colorado: Sheridan School District 2 (49 percent) and Littleton School District 6 (9 percent)
• Illinois: Carbon Cliff Barstow School District (45 percent) and Geneseo Community Unit School District 228 (6 percent)

Welcome to How I Teach, where we ask great teachers how they approach their jobs — and what's inspiring them outside of school. You can see other pieces in this series here.

More than a decade ago, Jonelle Hinchcliffe traded a career in advertising to become a New York City teacher. She hasn’t looked back.

Now, Hinchcliffe teaches geometry to high school students at Westchester Square Academy in the Bronx using the “flipped” model. In a flipped classroom, students are expected to spend time at home watching videos that introduce key concepts, and class time is then devoted to critical thinking or group exercises.

She won a Big Apple Award and the Empire State Excellence in Teaching Award for her work this year. She also participates in the city’s model teacher program, opening up her classroom to other educators.

What's a word or short phrase to describe your teaching style?

Engaging. Collaborative.

What’s your routine like when you arrive at school?

Rushed! Greet colleagues, organize room, set up ‘do nows,’ pass out student folders, and open up the room for students who need to watch last night’s video. I greet students at the door five minutes before the first period bell rings so we can start early.

What does your classroom look like?

Colorful! Artsy. I like to teach geometry through art projects that use geometry theorems, constructions and tools. All our artwork hangs on the walls.

What apps/software/tools can’t you teach without? Why?

ShowMe. I use it to create videos every night for my flipped class. My students don’t get homework. Instead, they watch a video I record in preparation for tomorrow’s class. They take notes on the video and then come into class ready to practice their new ideas. There is no direct instruction in class.

How do you plan your lessons?

I look at the standards and organize them into a curriculum map. (This is fairly easy in math, since ideas are sequential and build upon each other.) Then I group topics into units that are about three or four weeks long. I wrap proofs into every unit.

Daily lessons surround a topic or theorem that I want kids to understand. Since they are exposed to the idea the night before in my video, my ‘do now’ must bridge that content into the new work. Two problems from the ‘do now,’ and we’re ready to collaborate into the day’s work.

What makes an ideal lesson?

I know I have planned an ideal lesson when the majority of the room understands it, and I can get to every other student in the room to explain it if they don’t understand.

Also, a lesson has flowed perfectly if I provided all the appropriate steps and information in the video the night before. Then they have that information in their notes, so they can help themselves in class!

How do you respond when a student doesn’t understand something?

I sit down with them and find another way to explain it. I have little white boards that I carry around the room, and we use those to re-explain and to have kids practice a bunch of times until they get it.

What’s your go-to trick to re-engage a student who has lost focus?

Again, I sit with them. If I have to, I’ll hold the pencil while they tell me what to write. Eventually, when we get our focus back and energy redirected, I’ll pass the pencil back and stay with them until they really seem reengaged.

How do you communicate with parents?

I email every one of my parents every night to let them know there’s a new video. They love it. I also email them every time there is a test or project due, and to remind them about parent teacher conferences. If email is insufficient, I will call home to keep them informed.

What hacks do you use to grade papers?

Scantron sheets! Swap-and-grades, where kids grade each other’s papers, too.

What are you reading for fun?

The last book I read for fun was "The Selfish Gene" by Richard Dawkins.

What’s the best advice you ever received?

Take the job seriously, but don’t take yourself too seriously.

This is the second entry in a new series we're calling How We Got Here, where students and families explain how they chose, or ended up at, the schools they did. You can see the whole series here.

Even before my children were born, I spent hours envisioning them in a kindergarten classroom, smelling of crayons, Tempera paint, and Elmer’s glue. Little scissors, blocks, and dress-up clothes completed the Norman Rockwell picture.

Once my son was born, I spent years thinking about how to move to an area with a good school. We worked hard and eventually bought our way into Brooklyn Heights, a sought-after neighborhood that is home to the coveted, and overcrowded, P.S. 8. By the time I filled out my son’s kindergarten application, it was a moment more than five years in the making.

I waited with anticipation to receive our acceptance letter. When it came, I opened the envelope and read, “We are pleased to announce that your son will be attending P.S. 307.”

Waves of panic washed over me. P.S. 307, while also nearby, is not P.S. 8. Most of its students live in poverty. I sat awake at night envisioning my precious little blond boy going to a school with a towering housing project in its shadow. In that moment, I wondered: How could this happen?

Months later, I wonder more about how I could I have reacted that way. I want to take you on my journey as I faced my own ignorance — as “choosing” a school taught me about my own prejudice.

During those first panicked weeks after receiving the letter, I desperately looked for other options for my child. Then I got a phone call with a cheery female voice on the other line. “We are inviting all of the families who were accepted to P.S. 307 to the school this Saturday to welcome you,” she said. “Would you be able to attend?”

And so I went. When I arrived, a friendly security officer said hello. I heard trickling water and turned to see two handsome turtles in an aquarium.

I learned that the turtles are a result of a recent grant to boost science education. The school has on-site STEM director who helps the teachers write highly engaging, rigorous, hands-on, project-based units connecting science, technology, and math. Each class has an animal, insect, or plant in the classroom. Literacy lessons connect to the science content.

I also learned how much Principal Roberta Davenport cares about her students. She grew up in the public housing across the street, and drove in every day from Connecticut to serve the community where she grew up.

When I visited again later, students were on task and engaged. It was a quiet and peaceful environment. A pre-kindergarten class was testing out vehicles in the hallway that they had created in the science lab. One class was singing as a teacher played guitar, while was reading and playing music on keyboards.

The pre-K classrooms were inviting, with lots of nooks and cozy corners. I learned that the pre-K and kindergarten classes get 20 minutes of playground time in addition to regular recess. I learned that the school has a band, a chorus, and even offers violin lessons.

I was stunned by this little school, which I had so harshly judged. I felt energized and full of hope for my son, and I was angry with myself for making assumptions that this school would be unimpressive because it served poor students.

But while I felt hopeful, the other parents on the tour with me had more negative reactions.

“Why are test scores are so bad? Explain that to me.”

“It is just such a long walk to the school. It is too far.”

“The kids have to wait outside in the rain when they get picked up. I can’t have my nanny wait in the snow and rain like that.”

I felt like these parents were finding reasons to discredit the school before learning more. Why couldn’t they see what I was seeing?

This started my effort to try to understand what we were all looking for. What messages have we internalized? How do those ideas come out in ugly ways when our children are at stake?

One parent told me she has to consider the people her child will meet in school. “He could make some great connections that could help him later on,” she said. I guessed that she meant that she wants him to get a good job. I also think she meant that she wants him to be part of upper middle-class society.

I remember another conversation with a friend over a cup of coffee in Brooklyn Heights. I asked, what kind of school do you want your child to go to? It was a heated discussion with lots of words dropped: safe, nurturing, inquiry, project-based.

The more I dug into my soul, the more I come back to the idea that choosing a school is essentially about choosing the class you want to belong to and not about education at all. The truth that nobody wants to talk about is they want their white child to go to a white school. They want to go to a school where everyone eats Pirate Booty instead of Cheetos, where parents send their kids to New York Kids Club instead of Chuck E. Cheese’s.

I bring my own desires and biases to this, too. I am an educated, white, privileged woman. I am also a teacher.

I know from experience in my classrooms that upper-middle-class kids who come from educated backgrounds do well. They test well. They go to college. They get jobs. They do fine. As a teacher, I’ve also learned that the more diverse a classroom, the richer the discussions are, the more empathy is grown, the deeper we go into content knowledge.

And yet, even after deeply thinking about this — even after acknowledging these impulses — I did not happily send my child to P.S. 307. It was a complicated decision that followed many fraught conversations.

Ultimately, I decided that I wasn’t ready to take such a dramatic step. I wanted my son to have an educational environment like the one I saw at P.S. 307, but I couldn’t get past the idea that the level of need that most of its current students had was too different from my child’s. I also worried that, with Principal Davenport leaving, I wouldn’t be able to count on the quality I saw being sustained. And maybe the final decision was also rooted in a deep need to belong in my own whiteness.

But my son’s offer to P.S. 307, and my months of thinking on this issue, also led me somewhere surprising. Just a week before school began, my son did receive a seat at P.S. 8 — my dream scenario from the start.

We declined it. I knew that going to P.S. 8 would only further the institutionalized segregation in the city. I had become too uncomfortable with what the school’s relative homogeneity would mean for my son, despite the school’s enviable partnerships with Lincoln Center and the Guggenheim Museum.

Instead, I chose to send my son to P.S. 261 in Boerum Hill, a school with a more even mix of white students and students of color. It felt like a school where my son would be exposed to classmates truly different from him, but without the worries I couldn’t shake about P.S. 307. There, he could choose between chess or double dutch, gardening or African drumming, ballet/tap or hip-hop dance. It felt right.

The online review of P.S. 261 notes that “It has an active parent body that includes lawyers, hairdressers, writers and maintenance workers. This frank-talking community embraces the friction ethnic and economic diversity can sometimes bring, believing that kids coming together from different backgrounds creates a better world.”

This feels like a glimmer of hope. When we push past our neighborhood and visit each other’s schools, we can see into another world of possibility. And maybe when we do, we start to block those covert messages that are sent to us when we only attend white, upper-class schools.

Maybe those messages are born from segregation. And just maybe, sending my son to a more integrated school can start to break the cycle.

Interested in contributing to How We Got Here? Email us.