What’s Up With The Math Revolution

The Atlantic has a new piece about the more and more advanced work now studied by middle school and high school students.  This trend is well recognized among those of us who’ve been working with these students for years, although it was interesting to read about it in a publication intended for a general audience!

The article is largely about math beyond what’s typically taught in school.  You might call it high-level math, more challenging math, more abstract math, or maybe STEM Pathway math because it helps kids develop the skills to succeed on that pathway.  Regardless, it basically doesn’t touch at all on what a school math curriculum should be or what math class should be like.  It’s important to keep that in mind; this is math for kids who want more math.

And let me say, it’s wonderful that kids have so much opportunity to challenge themselves with interesting mathematics.  However, there are two big downsides to the current arrangement.  First, some students may feel pressure to advance themselves for the wrong reasons.  Second, those without access are being left ever further behind the top achievers.

I’m honored that BEAM was discussed in the article as an effort to address inequity in access, and that I get to contribute to this conversation, but we’re still quite a ways away from really closing the gap.  In this post, I want to dive into what’s really going on so that we can understand how the next generation of mathematicians and scientists is growing up.

Continue reading “What’s Up With The Math Revolution”

An Actual Response to Chief Justice Roberts

On December 9, the Supreme Court heard oral arguments (uhhh, again) in Fisher v. University of Texas, a case about affirmative action.  At issue: what measures (if any) can the university use to increase diversity if those measures disadvantage White students?

The case provoked all the expected outrage, especially when Justice Scalia made a half-informed comment about the effects of affirmative action on minority students’ ultimate achievement.  However, while listening to back episodes of the Amicus Podcast, I heard a different comment from Chief Justice John Roberts.

“What unique perspective,” he asked, “does a minority student bring to a physics class?”

The lawyer for the University of Texas, not surprisingly, was tongue-tied.  (Not exactly part of prep for the case, eh?)  A casual internet search revealed many non-response responses explaining why diversity is important and why physics needs underrepresented students to succeed.  (That second link, if you’re curious, is a letter from almost 2500 physicists to the Supreme Court.)  The Atlantic has a lovely piece about Einstein’s journey to discovering relativity and how it relied on philosophy, but the piece still could only hint at an answer to Roberts.  Somehow, none of these responses actually answered the question!

That’s where I’m stepping in.

Unique Perspective #1: Communication

The job of a physicist is centered around two things: making new scientific discoveries and communicating those discoveries.  A discovery that is not communicated is useless.  Physicists write up their work in academic journals and give talks at conferences.  For many of them, the bulk of their academic employment will be based around teaching physics classes.  Those who go into industry must communicate with coworkers, management, and the public on a regular basis.

Successful communication requires being able to phrase your work in a way that can be understood by those of many different backgrounds.  In lab settings, in group projects, in presentations, it provides a key benefit to learn how to communicate with those who don’t share your background.

Unique Perspective #2: Applications

Many people taking physics classes are going on to think about applications of their work to the real world.  Perhaps they are engineers and will be building bridges.  Perhaps they are going to work at NASA or SpaceX or Blue Origin and will lead space exploration.  Perhaps they are going to work in nanotech, or semiconductors, or… you get the idea.

In all of these cases, applications to the real world are essential.  They must design technologies to be used by other people.  They must think about how the bridges they build interact with the communities around those bridges.  Diverse perspectives allow students to better understand the applications of their work, how it will be used, and how to design it for maximal benefit to society.

Unique Perspective #3: Cultural Support

For the sake of argument, let’s suppose that there’s a physics student who gets into UT and she’s be the only Black student in her class.  She’s doubly underrepresented: one of few women, and the only Black student.  Her learning will be negatively impacted because she has no one to talk to about those struggles.  There’s no one who can understand the lack of role models or the biases she faces.  If she comes from an environment that is not middle- or upper-class, there is no one with whom she can discuss the culture shock not just of attending the university, but of physics, which has its own cultural norms.

This student, although admitted on her own merits, is getting an inferior education to others because she does not have a supportive peer group.  This is preventing her successful education, because her class lacks the perspectives of other students that will help her succeed.  Without diversity, the University of Texas cannot do its job for her, cannot give her the service for which she is paying tuition.

Concluding Notes

These are not the only reasons I support affirmative action in educational settings.  However, as someone who has designed numerous educational programs in math and science settings, I have sought diversity of viewpoints and backgrounds not for a social justice purpose, but because that is how I can provide the best educations for my students and create the products that students will want.  As the country becomes more diverse and as students enter a globally competitive marketplace, access to diverse viewpoints is an essential part of a good education.

To put it in the starkest terms, denying the University of Texas the tools to create a diverse class will decrease their educational effectiveness and put them at a competitive disadvantage against other educational options that offer greater diversity.

BEAM 6: Designing a Schedule, or How Does It All Fit?

This is part of a sequence of posts developing a new project for Bridge to Enter Advanced Mathematics called BEAM 6.  BEAM 6 will be a non-residential, four-week summer program for underserved 6th grade students in New York City.  You can find the other posts about its design here.

How can we possibly make everything fit?

Seriously, we have a highly ambitious curriculum planned and we must also fit a vibrant social schedule. The community formed at BEAM 6 will carry students through their future studies if we get this right; it will provide a beacon that tells them that they can have good friends and be serious about math at the same time. It will also encourage them to continue on to BEAM 7, and we want them to come! What schedule will let us accomplish these goals?

The Basics

BEAM 6 will run six days per week. Five days will be class days, which will mix both classes and activities. The sixth day will be a field trip or activity of some kind to further build community and friendships.

We need to fit in as much time during the week as possible, but we have a serious limitation: rush hour! Students will be brought to the program by our undergraduate counselors, who will meet them at subway stations near their homes. However, navigating the subway during rush hour with a bunch of 11-year-olds is not a great plan.

We can’t avoid rush hour completely without terribly shortening our day, but we can avoid the worst of it. If we start at 8:15am, then the farther students will be boarding the subway at 7:15am, which is not too bad, and by avoiding an “on the hour” start we also avoid peak times. Then if we end at 3:40pm, we can get on the subway before the afternoon commute.

That’s our day: 8:15am-3:40pm. I’m not crazy about how that schedule makes us feel like a normal school, but we have few other options.

I debated for a while which day should be our sixth day. BEAM 7 runs on a Tuesday-Saturday academic schedule (adopted from Canada/USA Mathcamp), with field trips on Sunday and Monday. The advantage of Monday trips is that you can visit when places are almost entirely empty. However, such a schedule might cause significant confusion among students and families, and the subway has unpredictable problems on weekends that could interfere with students’ on-time arrivals. So, in another compromise, classes will run Monday-Friday, with activities on Saturday. We might reconsider this in future years when we have more capacity to deal with any unexpected challenges an unusual schedule causes.

Program-long Schedule

We have five courses planned: Logical Reasoning, Applied Math, Math Foundations, Math Team Strategies, and Exploring Math. We cannot possibly offer all four in the same day. With lunch, two activities, and study hall, we simply run out of hours.  (Study hall is very important to me, because it gives students time to reflect on their work and instills study habits.)

The first decision I made was that “Exploring Math” can naturally be simply “Afternoon Math Circle”. The last thing in the day, it’s a fun piece of math, different every day, taught by different people. We can also use the time for guest speakers talking about how they use math in their work. Regardless, this will be in the final block, 2:40pm-3:40pm.

For the rest of the courses… well, let’s consider two different options.

Plan #1: Two two-week sessions

The four other courses naturally break up into two groups of two, so we can have 2 two-week sessions. Students would focus on one pair of courses during each two-week session.

Applied Math with Math Foundations. I paired these because Applied Math will likely be the most intrinsically exciting course, and Math Foundations the least exciting. Applied Math needs as much time as it can get: if students are to become independent in programming, then they need to do lots of programming. Hence, while Applied Math can expect to give 1-1.5 hours of homework per day (depending on the day – see below), Math Foundations should give no more than 15-30 minutes per day. Math Foundations is not designed to drill students in procedures, but rather to encourage creative solutions to problems, so it is all right to give less homework.

Logical Reasoning with Math Team Strategies. Math Team Strategies would get the bulk of the homework time, because we want students to become acclimated to using online resources such as Alcumus and the Art of Problem Solving forums.  These classes are both in the middle in terms of intrinsic excitement, so they pair well together.  Additionally, by putting Math Foundations and Math Team Strategies in different sessions, we know that students are always getting something related to math they’ve learned in school.

Plan #2: Odd/Even Days

There is another way to divide the courses. Instead of two groups of two, they could alternate days. One day is Applied Math and Math Foundations courses; the next day is Logical Reasoning and Math Team Strategies. In this plan, all courses run the full four weeks, but every-other-day.

Pros, Cons, and Choosing a Schedule

With Plan #1 (2 two-week sessions), it is easier to find faculty (who can now teach for just two weeks) and students can focus on specific topics as they go along.  Moreover, by studying the same thing each day, teachers don’t have to spend as much time reviewing at the start of class.

On the other hand, with Plan #2 (alternating days), students get more practice balancing competing demands on their time with homework assignments. Moreover, their ultimate recall is stronger because they spend a longer time actively engaged with each topic.  Finally, it makes things more uniform.  For example, suppose that in Plan #1, someone is teaching Math Team Strategies.  For the first two weeks, their students are new to the program and haven’t taken our Math Foundations course yet, so they will struggle.  But when the course is repeated for different students in the latter two weeks, all of those students have had Math Foundations.  If we use Plan #2, this goes away.

After talking with my colleagues, we settled on Plan #2.  We feel that it is a better educational experience for the students.  While finding faculty may be harder, it is worth it for a stronger program.

Daily Schedule

At BEAM 7, the courses provide no homework. Students do all their work in class, with attention from the instructor. This allows for a fast-paced, highly-interactive environment. However, there are disadvantages as well. It doesn’t train students to budget their own time and develop independent work skills. Moreover, it doesn’t fit well with part-time faculty for a day program. At BEAM 6, we’ll have shorter classes and time for students to do work.

My first draft of the schedule came out like this:

8:15am-8:30am: Breakfast
8:30am-9:30am: Class
9:35am-10:35am: Class
10:40am-11:40am: Activity
11:45pm-12:15pm: Lunch
12:20pm-1:30pm: Study Hall
1:35pm-2:35pm: Activity
2:40pm-3:40pm: Afternoon Math Circle

Lunch can be short, because we will almost certainly get catered boxed lunches that students can grab and eat. Since it is right after activity, it still provides a good break from their classes. If we had just one more hour, I could fit two hours of class/study hall/whatever between lunch and activity, but with avoiding rush hour we just don’t have that time.

However, after reflecting on this schedule, I want more time for study hall. Especially for the programming course, there just isn’t much time for student independent work. Currently students would have a total of 20 hours of work on programming (10 with the instructor and 10 in study hall); more time would be a huge asset. Moreover, having 20 Afternoon Math Circle sessions, while delightful, is not really necessary. Hence, on some days we can replace Afternoon Math Circle with a second Study Hall time. In the end, I decided that Monday, Tuesday, and Friday will have Math Circle (good way to end the week!), while Wednesday and Thursday will have extra Study Hall, allowing students to work on projects or longer assignments later in the week when they are in the thick of things.

This is all very complicated.  Now we have odd/even days determine which of the four long-running classes are happening, while days of the week determine if Math Circle is happening.  I think these are all the right decisions, but we will need clear messaging to make it work and make sure that students feel comfortable with their schedule.

Other Times

There are, of course, a whole wealth of other details.  For example, at what point do students select their courses?  Should they do so on the first day of the program (which eats up class time), or in some earlier orientation?  Right now, my plan is to schedule an orientation for students and families before the first day to talk about the program and how it will work, and to include course selection there.  Unfortunately, some students will miss that event, and we will have to give them another time for course selection.

We must also schedule Saturday trips.  We are thinking about movies, or a trip to the Bronx Zoo, or similar events.  These will each have their own schedule based on what we are doing.

We may also want some sort of closing ceremonies with parents.  Again that will require separate scheduling.  Most likely, we will go for Friday night after the program is done, and provide some sort of food.

Finally, we must have training/setup and wrap-up/take-down with staff.  I am planning the Friday before the program for the former (full-day for counselors, half-day for faculty) and the Saturday after the program for the latter.

I’m sure there are other details that we will think of as we go along.

Wrapping Up

Things fit.  They don’t fit as much as I want; the day feels too short to me, making it hard to really bond with everyone as much and get as involved in the classes.  I am worried that we won’t be able to instill in students the habits we want them to have for their educations.  But this is an iterative development process.  We will run a great program, and then make it even better for next year.

Despite any shortcomings in our available time, this will be a tremendous experience for students.  It will open up so many educational pathways.  Seeing a concrete schedule really gets me excited for the summer!

How hard should it be to pass algebra?

New York State is grappling with the difficulty of the Common Core Algebra test.  The intent is to raise the passing score to require real mastery of the material, but realistically speaking, most students are not reaching mastery.  (In fact, even with the original, very low standards, some students had to take the exam many times to pass or might not pass at all.)

The core of this issue seems to be: what is the purpose of teaching algebra?  For example:

  • If the purpose is to preserve the opportunity for all students to enter science/engineering/math, then the standard should be high.  It does no good to a student to barely scrape through algebra if they want to be a scientist.
  • If the purpose is to give everyone exposure to a beautiful subject, then the standard should be kept relatively low: it is the exposure, not mastery, that is important.
  • If the purpose is to give people access to math they need for life, then algebra should be dropped or revamped.  Many people do not need algebra in life, and a high barrier to graduation does them no good.

Right now, the grade required to pass is being used as a proxy for this kind of battle.  Those whose focus is on high school graduation want the required grade to drop.  Those whose focus is on preparing students for STEM careers want it to go up.  Without resolving this difference of goals, everyone will just keep shouting at everyone else and we’ll end up with a muddled policy that drags students in multiple directions.

Alas, that is not so unusual.

NYC’s high school selection system might be the worst system for choosing a high school — except for all the others

The New York Post published an opinion piece about the byzantine high school selection system.  As someone who’s helped well over a hundred students through this system, it really is byzantine.  There are over four hundred non-charter public high schools in the city, plus dozens of charter high schools.  In eighth grade, students can complete a “Round 1” form where they rank the high schools they are most interested in.  Each of those schools have different criteria for judging admission: some are based on your grades, test scores, or attendance; others, on the level of interest you show by attending information sessions; others, on a special test you take for admission; others on a portfolio you submit; others on an interview you do; others on geography; and finally many combine several of the above.  Just as you rank the schools, the schools rank you.  Then an algorithm is done to match students with schools.  Just imagine having to research all those schools, understand each one’s admissions priorities, and then complete all those applications!

Sound complicated?  We’re not done yet.  In a separate process, there are the specialized high schools (think Stuyvesant, Bronx Science, etc.) which have their own test, the SHSAT, which you take for admission.  Admission is based solely on this one test, and many people study for years.  (A sad waste of talent.)

No, we’re still not done yet.  Based on the SHSAT and your Round 1 form, you get a school assignment in March, which could have 2 schools (if you got into one of each, you get to choose), 1 school (if you got into one “normal” school and one specialized school), or 0 schools (if you were not placed with any of your Round 1 schools nor did you score high enough on the SHSAT).  At this point, you can complete the Round 2 form, in which the same matching process is used as Round 1, but with those schools that are left over.  (Usually, not the good ones.)

If the Round 2 process doesn’t work out, you are automatically placed into a local school that has space.  But it’s still not over.  You can appeal your choice if something went wrong.  You can also apply separately to charter schools (yes, they’re in another totally separate system), which are decided by random lottery, but the more you apply to, the better your chances.

All of which is to say, the process is truly, incredibly complicated.  (I also simplified it, leaving out LaGuardia School of the Arts, private schools, and many intricacies of appealing your placement.)  As the Post‘s opinion piece justly points out, this process dramatically favors more affluent students who have much better coaching and access to information.  (Let alone that they often speak better English.)

You might think, at this point, that I agree with the author that we should abolish this system.  After all, it clearly favors affluent students and takes a lot of time for everyone.  But there is one key problem: nothing better has been proposed.  At least the current system gives students a chance to go to a better school.  At BEAM, we can coach students on how to get into a great school that will challenge them, and we have tremendous success.  Some families find their way on their own.  Compare to a system that just places students by geography, in a city that is highly segregated (by race and income) — what chance would low-income students have then for access to these schools?

This is a subtle issue.  Maybe a system based only on geography would help, because then there might be some mixed-income schools (although I find that unlikely).  Maybe the current system has another downside, “creaming” the best students from low-performing schools, leaving them worse off.  These are interesting questions, questions that deserve study.  But it’s not worth changing a system that offers some kids incredible opportunities unless you’ve done a very careful examination of the trade-offs, and we frankly have no idea!

So what should we do?  If more students had real advising on navigating the high school system, they could be vastly more successful at getting into great schools.  There are so many big mistakes that students make all the time: not filling in all 12 spots on the Round 1 form, for example, or incorrectly judging what school to apply to.  (We had a student who decided by looking at schools’ graduation rates, not realizing he’d ranked a dual-language school specifically for English Language Learners, a decision that he is now stuck with for 9th grade even though he does not speak fluent Spanish!)  Just as we need more guidance counselors for college applications, help with the high school application process might create a system that offers tremendous opportunities for all students, regardless of background.

Algebra for All in NYC

Mayor Bill de Blasio has announced a plan for every New York City middle school student to have access to algebra in 8th grade, and Chalkbeat is out with a great story discussing the challenges facing the initiative.

I work with many exceptional middle school students at BEAM, and for many, the lack of algebra in their middle schools is a tragedy.  No access to algebra closes doors; it’s as simple as that.  That’s because most people who pursue science degrees have taken calculus in high school, so you’ll be significantly behind if you don’t have that background.  But getting to calculus in high school requires taking algebra in 8th grade, unless you double up on math in some year!  There are ripple effects throughout high school and college; no 8th grade algebra makes it much harder to pursue a science major in college (which is why BEAM offers an online algebra course).

Yet simply providing some algebra classes opens a host of other issues.  First of all, algebra must be taught well, which will be hard for teachers who are not experienced teaching high school level math and who may not have the mathematical background to offer a high quality course.  Moreover, in many schools a teacher will have several classes of normal 8th grade math, and one algebra class—where would you put your prep time?  But the bigger danger is that kids who are not ready for algebra will be pushed into it.

The Chalkbeat article seems to treat this as a good thing, talking about “lower-performing students who could use the early exposure to a subject that trips up many students in high school.”  But if a student does not deeply understand earlier material, then accelerating is a mistake.  A study in California (also, paradoxically, in the same Chalkbeat article) points out that when California mandated algebra in 8th grade, those students got lower scores on 10th grade math two years later.  Really understanding algebra requires really understanding math.  Otherwise, you’re just memorizing formulas and not learning anything.

In fact, algebra is something of a debacle right now in New York.  On the old, easier, pre-Common Core Regents exam in New York City, roughly a third of students had to take the test multiple times to pass.  (And let me tell you, a pass does not demonstrate mastery.)  A quarter of students had to take it at least three times.  There were 66 students in New York City who took the test ten or more times, and half of them still didn’t pass!  Even for those students that did pass, did doing so on the tenth try really mean that they got something out of algebra class?

Fundamentally, there is a mismatch between what we are aiming for, namely that all students take algebra, and what we are achieving, which is that many students take the test repeatedly, focus on memorization, and don’t learn mathematics on a deep level.  Algebra should open doors.  There are no doors being opened by memorizing formulas, which (barring exceptional teachers) is all you can do when you don’t have the mathematical thought processes down going into the class.

It is good that algebra will be available to everyone.  It is a critical equity issue.  But for students to really take advantage of this opportunity, there is a lot more groundwork that must be put in place.