Five W Questions
AR: Thanks very much, Amy, for agreeing to be interviewed for the Journal of Statistics and Data Science Education. Let’s start with three W questions: Where do you teach, what do you teach, and who do you teach? I’ll ask about why and when later.
AH: I teach at Brooklyn Technical High School. Brooklyn Tech is one of the specialized public high schools in New York City for which students must take an entrance exam. We have about 6,000 students coming from all five boroughs. I teach AP Statistics and a Math Analysis class, an elective course with extracurricular math topics for our sophomore math team members. Some of my AP Stats students take it as a required class for their major and some take it as a math elective.
AR: How many students apply to your school, and what’s the acceptance rate? Is acceptance based solely on performance on the entrance exam? What does the exam assess?
AH: I am not sure about how many apply or the exact acceptance rate. Right now acceptance is based solely on a student’s score on the Specialized High Schools Admissions Test (SHSAT), which has English and math sections similar to the SAT. I believe last year around 1800 students were given offers to our school.
AR: Just as I was not familiar with specialized public high schools, I’m also unfamiliar with the idea that high school students major in a subject. At what point do students declare a major? What courses comprise a math major?
AH: Majors in high school are not common, for sure. The major system at our school is one of things that really sets it apart from other high schools. Similar to college, sophomores choose their majors at the end of the year with most students getting into their first choice. The math major juniors study graph theory, number theory, and write math research papers based on original work from a topic of their choice, a really phenomenal opportunity for some wonderful mathematical thinking. The seniors take AP Calculus or Multivariable Calculus, AP Statistics, AP Macroeconomics, and Linear Algebra. The Finance majors also take AP Statistics their senior year. Overall, I would say roughly two-thirds of my students are taking AP Statistics as a course required for their major. That means about one-third are juniors and seniors in other majors taking it as a math elective. It’s a great mix of students.
AR: It sounds like you have very well-prepared and high-achieving students. When did you start teaching there, and is this your first teaching position?
AH: I started at Brooklyn Tech in 2007. Prior to that I had been teaching at another public high school in the Bronx. Technically it was two schools, but they were in the same building and much of the staff had worked at both. The first, Columbus High School, was broken up into several small schools, and I was rehired at one them, Collegiate Institute for Math and Science.
AR: Now for the hardest of my W questions: Why did you decide on a career as a math teacher?
AH: When I was studying math as an undergraduate, I didn’t consider math education as a career for myself. In fact, it was something I adamantly said I would never do, maybe as a reaction to having a mother who was a professor. For a while, I was an actuarial analyst for a private benefits consulting company in Boston. After September 11th, I began to question whether or not I wanted to stay in Boston and, more importantly, if I could see myself doing actuarial work long-term. Although I liked the computational aspects of my job, I was feeling a little unmotivated by the structure of the work: the exams, fiduciary liability, and business aspects of it.
At the time, I was training one of my colleagues for the specific type of pension funds I worked with, and I recall mentioning how much I enjoyed the training to a friend who was teaching in New York City. She encouraged me to apply to the NYC Teaching Fellows, an alternative certification program. It seemed like exactly the right thing for me. I remember making up my mind about pursuing teaching, putting in my notice at my company, applying for the program, and moving to New York within weeks.
Once I began teaching, I realized there was a very unique satisfaction with getting to do and think about math and statistics all day with students.
2020
AR: Speaking of uniqueness, let’s talk about your experiences in the year 2020. Please tell us about your last face-to-face meetings with students and your transition to teaching remotely.
AH: My last day with my students was Friday, March 13th. There hadn’t been an announcement yet for school building closures, but the reality of the pandemic had already set in here in NYC. I knew a building closure was inevitable, and I was trying to get things ready quickly for the possibility of remote instruction. I had already set up an assignment for students to practice uploading work electronically. I had already taken inventory of what parts of the curriculum were left to teach. That last day, I talked about a mile a minute as I reviewed an in-class activity, because I knew that I potentially had a very short window of time. The announcement was made the following Sunday to close the buildings. We had a “snow day” the following day, with no school. Ultimately, we had the next three days to work on transitioning our classrooms to an online platform. It was very rushed, very stressful. In the meantime, we were dealing with the reality of the growing number of COVID-19 cases in our community and in the city.
AR: How did you meet with your students and continue your teaching for the rest of the term—synchronously or asynchronously or in some other way?
AH: Any synchronous instruction I did was optional. I did virtually all of the remote teaching asynchronously for a bunch of reasons. First, I considered the equity issues that transitioning to remote learning had created. When students go to school, all students generally have access to the same classroom resources. As soon as we shift “school” to be in each student’s home, we are now creating potential differences in available resources. With my students, there was lot of variability with what they had at home to help them learn. Some of my students started without anything other than their cell phones to do their work. Some of my students live in large households where they are constantly sharing space, computers, and even childcare responsibilities as soon as the shelter-in-place took effect. So, I wanted to make sure that what I offered could be readily done by all my students, even those with limitations at home.
Second, teachers had been initially given permission to use Zoom, so I had my students practice with that platform even before we officially began remote teaching. But then the NYC Department of Education decided that it had some issues with Zoom, so we were told to use other programs. Later, the NYC DOE worked with Zoom to create our own version of the program and approved it again for teaching use. Ultimately, that meant some teachers worked with at least three different programs to deliver instruction. It was hard to plan with all these changes.
On top of that, there was a varied desire among my students to have synchronous instruction. In fact, when I surveyed my students about it, their responses had the most variation of any question I’ve ever asked on my end-of-course survey. Some really wanted it, but some were very happy being left alone by themselves to think through the material.
Ultimately, when making decisions about how I would approach remote teaching, I realized I would not be able to recreate my classroom experience with this remote setting and focused instead on finding and creating good resources to share with my students (including videos, presentations, and annotated student work) and making sure there was a clear work flow. I was available via email and video chat if students wanted to discuss the classwork. If I had to do remote teaching again, I would want to incorporate more synchronous aspects to have a good mix. I think that would be especially important for relationship building with the students and myself.
AR: How would you rate the effectiveness of this remote teaching and learning experience? What aspects (if any) of remote teaching might you want to keep, even after the happy day on which you can return to seeing your students in person on a daily basis?
AH: My students thought I did a good job adapting to remote learning. Or at least that’s what they told me. There are few, if any, aspects I’d like to keep when returning to face-to-face teaching. I already had been assigning short videos (calculator tutorials, for example) for students to watch at home, requiring students do some learning on their own, and using Google Classroom to share documents and make announcements even before this pandemic. If I had to pick one thing, it would be some changes with assessment. Our course team created a final “exam” that required students to write and answer their own statistical question as well as respond to some reflection prompts. It allowed quite a bit of flexibility and choice in what students presented as their summative work. I would absolutely do something like that in the future.
AR: That does sound worthwhile. Can you share some examples of what your students wrote about with that assessment?
AH: One of the content options asked students to write their own two-proportion z-test or two-sample t-test scenario, statistical question, and complete solution. Students’ contexts varied, and many used it to express their own personalities and interests: basketball, COVID-19, sleeping habits, social justice, and video games. I loved that some were serious reflections on current events (e.g., with Black Lives Matter), some were just goofy examples (like Energy Drink A vs. Energy Drink B), and one student even wrote a problem involving his favorite TV comedy that literally made me laugh out loud. We added three constraints just to add a little twist: a combined sample size of 250, all conditions and assumptions must be met, and the p-value had to be between 0.03 and 0.07. We have used a similar technique for a chi-square unit group project that’s a bit more involved: http://alittlestats.blogspot.com/2018/06/fake-data.html.
The one thing we added that we’ve never done before is to have students write two reflections (out of five choices), inspired by some prompts written by math professor Francis Su: https://www.francissu.com/post/7-exam-questions-for-a-pandemic-or-any-other-time. One of the most popular for the students to choose was to pick a career and write about a specific example of how a person in that career might use statistics to do their work. My students wrote about many interesting, mostly STEM-related, careers and examples. I think it was nice to see that my students didn’t have difficulty understanding that statistics is a useful tool for many different fields, everything from computer science to zoology.
AR: Just to make sure I understand: When you asked students for a complete solution, were they making up data to meet your requirements about sample size and p-value?
AH: Yes! To be fair, in almost all statistics problems, the actual numbers in the problem are very much one of the least important aspects about the problem. Having students make up the numbers (or, as we call it, intentionally fake the data) adds a very engaging mathematical and statistical challenge. Students have to demonstrate an understanding of both the mechanics of the distributions they’re using as well as the structure of the communication required to answer their own questions.
AR: I notice that you referred to your “course team,” and you have used “we” often. Please tell us about the collaboration among teachers at your school.
AH: I am very lucky to get to work with other wonderful math and stats teachers almost daily. Over a decade ago, I was the sole AP Statistics teacher in my building with two sections of students. Demand for the course has grown, and now we have more than 250 students taking the class. We had four different teachers teaching nine sections last year: Melanie Battles, Alexandra Brennan, Doug Shuman, and myself. The four of us work together regularly. We share a lot of materials with each other, and we have a couple of group projects that we plan together. This final exam was a group effort, and it was nice to make the final assessment the same for all the AP Stats students at our school, especially with remote instruction. Teacher collaboration is very much a part of our math department, mostly with course teams. In my not-so-humble opinion, though, our statistics course team is the best.
AP Statistics
AR: What’s your favorite topic to teach in AP Statistics, and why?
AH: I have grown to love teaching inference. At that point in the course, students have already gotten their feet wet with some of the tough statistical thinking, and I think it’s the first time many students appreciate the power of what they can do with what they’re learning. It seems fancy and grown-up while being applicable to so many things. Plus, I love that you can revisit so many of the other units organically and neatly. It makes writing class activities and questions so much more interesting.
AR: Please give an example of how you revisit earlier topics when teaching inference.
AH: I try to incorporate things like describing distributions from a graph, comparing two distributions, dependence and independence of variables, and experimental design. For example, you can conduct a quick in-class experiment and then analyze the data. In recent years, I have been collecting and comparing standing vs. seated pulse rates. The students figure out that a matched pairs design is ideal here. It’s very cool to show mathematically how matching does not result in independent variables and how much matching reduces variability in the response. I also give data for another comparative experiment, but one that is not so well-designed, and students have to analyze but also explain the limitations of the results.
AR: What topic in AP Statistics do your students find most challenging?
AH: Probability. I really have two kinds of students—those who look forward to the probability unit and those who don’t—but both groups of students find it to be challenging. Probability can be so counterintuitive, the math can get a little involved, and this is one of the areas where the graphing calculator (on tests, for example) is really clunky.
AR: How do you help students to approach probability questions?
AH: I try to de-emphasize reliance on formulas and focus on understanding the problem and organizing your solution. In fact, that’s my general approach for my math and statistics students. For basic probability, I love using tables and hypothesized frequencies. For the probability distributions, I try to highlight the math in a more conceptual way. For example, I have students explore Pascal’s triangle, which many of my students have never seen, and the binomial expansion to understand the binomial distribution, more to emphasize the binomial as a logical pattern rather than an arbitrary formula.
I also set up a communication routine for probability problems, so students have an algorithm to follow. That way, students always have a place to start and a set of questions they can ask themselves to try to get to the solution. What do we know? What do we want to find? What probability distribution can we use here? So many students think it’s just about the answer, but I actually think the real elegance is in how you can get to the answer.
AR: If you were given complete control over the AP Statistics curriculum, what topics (if any) would you remove and what topics (if any) would you add?
AH: I would probably remove nonlinear regression and add ANOVA. I love regression but I feel like with this course, we really can’t do it justice, and the non-linear regression unit seems a little out of place. On the other hand, I do think it’s good to expose students to some ANOVA as an extension of two-sample tests. Many former students tell me they appreciated that I cover ANOVA after the AP exam, because it’s something they see a lot in their college courses.
AR: Now suppose that you have complete control over the entire AP Statistics program, not just the curriculum. Would you make changes to anything else?
AH: I would eliminate the dependence on graphing calculators for tests, including the AP exam. Rote calculations don’t provide much interesting information about a student’s ability, in general. Perhaps all they convey is whether or not a student knew where and how to enter the information into their calculator. It’s probably a better use of testing time, then, to provide the values needed to answer a more interesting question. For the very first activity I assign, I provide all summary statistics and graphs, so students have to start thinking and explaining right away. In some cases, I give my students computer output and reference information to use for all their responses to questions. I think we saw with the remote 2020 AP exam administration that it is possible to create good questions that don’t require a graphing calculator, and I would like to see the program continue in that direction.
Math Analysis
AR: Please tell us about your Math Analysis class.
AH: We do all different kinds of math problems using the usual algebra, geometry, trigonometry, but also number theory, probability, and counting. We do math contests, discuss problem-solving strategies, and share ideas. It’s purely an elective class, and I get to spend time with my amazing sophomores doing math just for fun!
AR: How about giving us an example or two of a particularly fun or interesting problem that you like to give students in that class?
AH: So many fun problems… too many to choose from, but if I have to pick a couple, these are the two:
On the first day back from winter break I give my students a problem set highlighting the new year as a number. Math contests often have problems for which the year is part of a problem or part of the answer, so it really does come in handy, but it’s also just a lot of fun. I initially didn’t think 2020 problems would be that interesting, at least not in comparison to some of the ones from 2019, but in the end there were quite a few that my students enjoyed. Out of the 20 problems, the one that my students liked thinking about the most was this one:
Can you create an expression for which the result is 2020 using…
… all positive digits, in descending order, using basic operations?
… the numbers 1 through 10, in ascending order, using basic operations? Concatenation allowed.
… the numbers 1 through 10 in descending order? Same rules as above.
… the numbers 1 through 10 in descending order? No concatenation.
… each single positive digit (e.g., only 1’s)? Concatenation allowed. Is it possible for all positive digits?
… sums of powers of 2?
… sums of powers of 3?
… sums of powers of 4?
In remote learning, I posted a daily problem for my math team. I tried to curate problems that would be enriching and challenging, but also a good way for students to take their minds off current events. My students really liked the geometric-type puzzles that people like Ed Southall and Catriona Shearer post on Twitter. I think the overall favorite problems, though, were the Math Art Challenges from Annie Perkins. In particular, my students and I really enjoyed the Hitomezashi stitching challenge: https://arbitrarilyclose.com/2020/03/29/mathartchallenge-day-14-hitomezashi-stitching-suggested-by-katherine-seaton/. Some of my students, like me, did the challenge by hand and some even wrote computer programs for their responses.
Social Media
AR: You just mentioned Twitter and I understand that you are very active on social media. How has that improved your teaching?
AH: Social media can be a blessing and a curse. If you curate your follows well, though, it can be a very powerful teaching tool and resource. Twitter, for example, has had a positive impact on my teaching and on my classroom because I have the ability to be a part of a dynamic team of thoughtful and smart educators all over the world. I have been able to share ideas and resources. But, more importantly, the support network created by these online communities are crucial for teachers including myself.
I talk about this in my 2018 ICOTS paper, co-written with Lynette Hudiburgh: https://iase-web.org/icots/10/proceedings/pdfs/ICOTS10_C176.pdf?1531364321. Online support communities are especially important when teachers are the only one in their school teaching a specific subject, which is common with statistics teachers in secondary schools, and in times like these, during a pandemic. During these past few months, sharing the experience with other teachers provided me with a lot of comfort and insight into what has worked and what has not for remote teaching.
AR: As a social media novice myself, can you help me to understand what it means to curate your follows well? How do you go about doing that?
AH: Well, I think it’s easy to get carried away and think you have to follow everyone and read everything. That can be overwhelming and also not lead to a very interesting timeline. For my @alittlestats account, I like to follow a diverse group of educators, people and organizations posting about math and/or statistics, and those who post pertinent things related to my professional world. I want to read posts that are motivating, thought-provoking, and useful to my world today. I follow some hashtags, like #statschat and #mtbos (which stands for math twitter blogosphere), as well as have some lists for twitter feeds from certain types of people. For example, I have a list for people associated with Math for America.
AR: So many follow-up questions present themselves. Rather than be selective, I’ll just ask them all at once: About how many hours per week do you spend on twitter? Do you use other social media, such as Facebook or Instagram? Do you interact directly with your students over social media? And please tell us about Math for America and your connection to that group.
AH: The time I spend varies. The nice thing about social media is that it’s always there, but you can decide how much time you want to spend with it. I use other social media for personal use, but not for professional use. I like to keep those worlds separate for the most part. I do not use social media with my students. In fact, that’s not something my school would allow.
Math for America is a non-profit started by mathematician and former hedge fund manager Jim Simons. The organization’s purpose is to support and retain good math and science teachers in NYC public schools. Through my master teacher fellowship, I have access to an amazing group of like-minded educators, professional development, and additional funding to help support my teaching practice. The workshops they offer are mostly run by teachers and otherwise by incredible people in their fields. What I think what makes MfA so phenomenal is how much they as an organization trust and encourage teachers to be leaders and innovators in education, and to make a positive change beyond their own classrooms.
AR: Let me ask for two examples of things that have benefitted your teaching and therefore your students – one that you picked up from social media and one that you learned from Math for America.
AH: The math art challenges mentioned above were something I would not have found if it weren’t for social media. The level of engagement with my students with some of these was very high. I liked how the tasks allowed for a lot of creativity of thought.
As for Math for America, a favorite activity that I have used in my classroom is one we designed in our statistics professional learning team: a simulation for the 2016 presidential election. My classes’ results (based on a fairly simple simulation, picking random numbers) matched the published results from FiveThirtyEight within one percentage point. It was quite powerful to see students’ reactions when events they thought of as “unlikely” occurred.
AR: Can you describe more about how the election simulation activity worked? For example, was the goal to predict the outcome for the popular vote or the electoral college result, or both? What were some of the model assumptions behind the simulation? And how did you respond to students’ surprise at the 2016 presidential election result, which FiveThirtyEight gave a 28.6% chance of happening ( https://projects.fivethirtyeight.com/2016-election-forecast/)?
AH: The simulation is for possible electoral vote outcomes. We made assumptions galore to make it simple, including independence between states and ignoring the “safe bet” states (which of course our post-2016-selves know wasn’t the case). Repeat for each student and then compile the results. This is when it’s nice to have at least 100 students in your classes. I’ve been doing this simulation every year since 2016, and it’s always interesting to see how close we get to the professional results, as well as the variability in possible outcomes. Every year, we discover, yes, it is possible to have a tie in electoral votes. (And, of course, we flip a coin as a tiebreaker.)
I think the nice take-away for students is to really challenge their notions of likely vs. unlikely events. So many students previously had thought “likely” applies when there’s more than a 50% chance and, well, never consider 30% or even 10% to be pretty likely when it comes down to it. It’s a good foreshadowing activity to our study of formal statistical inference.
Pop Quiz
AR: Let’s move on to the “pop quiz” portion of this interview. I will ask several questions, some of which will move beyond the teaching of statistics, and I’ll ask that you keep your responses brief. First, please tell us about your family.
AH: My husband and I have celebrated our 14th wedding anniversary, but we’ve known each other since high school. We have parents, aunts, uncles, siblings, nephews, and lots of cousins all over the U.S. and abroad.
AR: What are some of your hobbies?
AH: Reading, puzzles, Pilates, and traveling (when we’re not in a pandemic).
AR: Please recommend a book for pleasure reading, a movie, a musical selection, and a place to visit.
AH: Any Agatha Christie book, The Italian Job (2003), Songs for the Deaf from Queens of the Stone Age, and Long Island’s North Fork.
AR: Here is a fanciful question: You can travel in time to observe what’s going on in the world for one day. What time would you travel to – in the past or the future – and why?
AH: Let’s fast forward to a time when I can finally hug my mom. I’m sure whatever is going on that day will be lovely.
AR: Let’s hope that day is in the near future. Here’s another silly one: You can have dinner anywhere in the world (all expenses paid) with three companions, but the dinner conversation must focus on teaching statistics. Who would you invite, and where would you eat?
AH: Dinner and drinks at The Rieger in Kansas City, please, with some fellow AP Stats readers and table leaders. (Don’t make me choose who to invite.)
AR: Let’s make the dinner list first-come, first-served from those who read this interview. Now let’s collect some data: Do you consider yourself an early bird or night owl? On what day of the week were you born? How many of the 50 states have you set foot in? How many miles do you live from your birthplace?
AH: Night owl (my pandemic self, though, has been waking up early). Born on a Thursday, 300 miles from here. I am up to 39 states (and hoping to get to all 50 by age 50).
Conclusions
AR: Thanks very much for your time and thoughtfulness in this interview, Amy. I have a few final questions in mind. First, you are the first recipient of a fellowship that the Statistics and Data Science Education section of the American Statistical Association has established. What is your role in that position?
AH: I am a liaison between the ASA and high school statistics educators like myself. I sit on the education section executive committee, give advice about how the ASA can be a resource for high school educators, and help get the word out about the ASA events that might be of interest to other K-12 statistics teachers. It’s been an interesting experience so far, and I’m still trying to figure out how best to contribute. I am in awe of the wonderful people on the committee and how dedicated they are to statistics education.
AR: For college professors reading this interview, what would you like them to know about the teaching of statistics in high schools?
AH: The role of a high school statistics teacher is not to teach the full canon of statistics. It’s to get students interested in statistics. Our job is more about letting students know what the possibilities of statistics are, and then their future college professors can fill in all the pieces. Teaching statistics to teen-agers is a wild ride, but it’s a good age to start because they are naturally curious and very often interested in a subject that helps them understand our complex world. I do hear from professors that my good students often become their great statistics students. So, you’re welcome.
AR: Do you recommend teaching statistics in high school as a career choice? For an undergraduate student who is considering this career path, what qualities should they have, or develop, in order to succeed in this role and enjoy it?
AH: Your question is a little tough to answer, because most high school statistics teachers are math teachers. It would be almost certain that someone teaching statistics would also teach other math classes, so you would have to acknowledge that there really is no such thing as a ‘high school statistics teacher’ as a career choice (with a few exceptions, of course). For high school math teachers who want to and know the content well, I highly recommend teaching statistics because it is challenging, engaging, and can include so many fun topics.
In general, though, I do not recommend teaching as a career for undergraduates, at least not as a first career. I know some people will think that’s harsh, but K-12 teaching is hard, even without a pandemic. It is especially challenging for someone new to the field. I think people interested in teaching math or statistics should take as many high-level math and statistics classes as possible, work in an interesting STEM field, and then … If they are still interested in teaching, make the transition after some work experience.
That said, I don’t regret switching careers. Teaching has been fulfilling in ways that I didn’t even anticipate. I think to be a successful teacher and really enjoy it requires good content knowledge, a way to develop a pedagogically-sound classroom routine, and the ability to reflect on what your students need.
AR: Thanks for that. Your response brings several follow-up questions to my mind: First, when you speak of what students need, are you referring only to their learning needs, or is part of your job to address their emotional and social and other needs as well? Second, what are some ways in which teaching has been rewarding beyond your expectations? Third, when you mention the keys to being a successful teacher, how do you tell when a teacher has succeeded?
AH: Yes, a good teacher teaches to the whole person in front of them. Teen-agers especially are bringing everything with them to your classroom. If you’re not at least acknowledging that somehow, there’s no way students are going to be able to think about your class content. I’m still learning how best to do that myself. Sometimes, maybe even most of the time, that’s the hardest part.
Hands down, the top reward of teaching is the kids. I get to spend my day with a group of young people who are funny, smart, kind, and full of personality. The bonus at my school is that I get to teach kids who, in general, like learning about math and statistics. I definitely didn’t expect to love my students as much as I do. I don’t think I could have stayed in this job if that wasn’t the case.
A successful teacher? That is the magic question, isn’t it? Measuring teacher success is something so-called experts are trying very hard to do. I don’t think of it as a quantity or even something completely tangible. But, at the end of the day, if learning has happened in my classroom, I consider that to be a success. When students are engaged, asking good questions, challenging their own thinking (and even mine), then that’s a great success.
Additional information
Notes on contributors
Amy Hogan
Amy Hogan teaches mathematics and statistics at Brooklyn Technical High School.
This interview took place via email on June 1–August 12, 2020.