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!

BEAM 6: Designing a Curriculum

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.

 

At BEAM 7 (a program many BEAM 6 students will attend the summer after 7th grade), we tend to throw students off the deep end when it comes to doing math.  I mean it: they come in and we teach them about proofs, and we have them solving MATHCOUNTS problems, and we have them learning number theory and combinatorics and even group theory… and a lot of them are kind-of still weak on fractions, y’know?

It varies by school, of course.  Not surprisingly, some schools tend to give us more prepared students, while others don’t.  BEAM 7 has had seventh graders (top of their class at school!) who were not comfortable multiplying negative numbers.

I’ve been asking myself what I wish our BEAM 7 students knew.  They’re held back constantly by foundational math knowledge.  They also need to learn how to look at a problem and focus on what it’s asking, rather than guessing at a solution mechanism.  Finally, I want them to have more skills in deductive reasoning and case analysis.  It’s a little bit crazy to be thrown into a proofs class without that!

In the end, there are five course tracks that I really want to work into the program.

  • Logic
  • Math Foundations
  • Math Team Training
  • Applied Math
  • Seminars

I want students to have choice, so each of these topic areas will have different courses within it.  During their summer, each student will take on course from each track.  The exception will be Seminars, where each will be independent (see below).

It’s a minor nightmare to fit all of these classes into a four-week program.  Right now, the best I can do is 10 hours of class time each, plus some homework time depending on the course.  So they have to be compact and get to the punch quite quickly.  Scheduling will be covered in detail in a future post, but lack of time is a huge concern.

Before I get to describing the course tracks, readers who know about BEAM 7’s courses will see right away that this is super different.  BEAM 7 is basically a playground for our faculty to develop all kinds of interesting math courses and then teach them.  These courses are much more structured and targeted.  Why?

The primary reason is simple: BEAM 6 has a very different goal.  BEAM 6’s main goal is to remedy specific gaps that students need to succeed both in BEAM 7 and in their future mathematical studies.  In contract, BEAM 7’s goal is to transition students to other programs for advanced study where they will have to do more abstract thinking.  Hence, BEAM 7 invites faculty to rock out in courses similar to what students will do at future programs.  In contrast, BEAM 6 is a laser aimed at skills and knowledge that students need.  BEAM 6 courses will be lots of fun, but they’ll also have much more concrete goals.

There are advantages and disadvantages to both approaches.  One big advantage of BEAM 6 is that I can develop a strong curriculum for students.  A second advantage is that it opens our program up to more potential instructors, because they do not need the same experience designing enrichment classes.  However, BEAM 6 is still open to those who want to create their own crazy classes through both the Seminars and Applied Math topic areas.

Great!  Let’s figure out what’s actually in the courses.

Continue reading “BEAM 6: Designing a Curriculum”

BEAM 6: Goals

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.

In about seven months, there will be 100 sixth-grade students all ready to learn math.  Almost exclusively, their mathematics educations so far will be designed around memorizing procedures and passing tests.  We have four weeks to change their lives.  What should we do?

No pressure or anything.

It’s rare in education to get an opportunity to work with motivated, talented students with no outside requirements.  We can design the program that is best for them.  This is the first post developing BEAM 6, and so we will set down the program goals.

Goal: Teach Them to Think Deeply

If students leave the program and they have not learned about logical reasoning, I will feel exceptionally disappointed.  I want students to grasp ideas of deductive reasoning.  This might be my single biggest goal.

I also want to change the way they think about mathematics.  For many students, math problems are defined by the solution method.  “Oh, this is an addition problem.”  “Oh, this is a related rates problem.”  “Oh, this is a Pythagorean theorem problem.”  This thinking leads to oversimplification and memorizing procedures.  It makes it more difficult to solve multi-step problems.  Students should engage with the question, understand the problem independent of its solution, and accept or reject solution paths because they do or don’t solve the problem.

This leads to the broader question of mathematical communication.  For example, the equals sign.  Students often interpret the equals sign as asking a question.  In elementary school, it is always used as “2 + 5 = ?”.  By algebra, the question changes — “2x – 3 = 15” means “solve for x” — but the equals sign is still primarily used to express a question.  Students don’t realize that “25 + 7 = 32” is a statement that can be true or false; that the purpose of = is not to ask a question but rather to give a statement.  The result is a failure of both communication and conceptualization.

These goals are less mathematically sophisticated than BEAM 7’s goals.  This is in part because the students are younger.  It’s also to build synergy with BEAM 7.  Students often come out of BEAM 7 with a strong grounding in abstract mathematics but still well behind peers in school-based math.  For example, students often do well taking a number theory course at CTY or going to a program like MathPath, but do relatively poorly in a contest like MATHCOUNTS.  BEAM 6 can close that gap and set students on a path to deepening their facility with school-based math.

Goal: Help Them Love Math

People love math because it is beautiful; because it is thrilling to challenge yourself with a hard problem that you finally solve; and because it is interesting to see how it applies to the real world.  We must show students what math really is.  That it is not about memorization or following procedures.  That it is beautiful and creative and exciting.  A love of math will carry you far, and we should develop it in the students.

Goal: Develop Their Self-Identities

In my experience, self-identity drives a lot about a person.  More than just thinking something is “cool,” self-identity can push someone to pursue an interest; it can create resilience to failure; it can drive life decisions.  If we can develop self-identities in our students as scholars, and furthermore as scientists and mathematicians, they are much more likely to succeed on that path.

What contributes to developing self-identity?  Here are some thoughts:

  • Interest/passion for a topic.
  • A feeling of self-efficacy; confidence in your abilities.
  • Membership in a distinctive community.
  • Role models.
  • A sense of future (where will it take you?).

We should harness all of these within the program.  We have special expertise in creating a mathematical community.  To drive students’ further engagement, creating a very strong community will be essential.

Goal: Develop Independent Learners

A summer program cannot alone cover the mathematical education of all these students.  If they will be successful, they must continue to pursue learning after the summer is done.

Students should be connected with resources for further study, such as Art of Problem Solving.  They should get used to these tools during the summer and be encouraged to continue using them when they’re done so that they continue to get better.

Concluding Thoughts

These goals feel right.  They cover what I feel is very important to develop in young mathematicians.  However, they are not complete.  While program elements will be tied into these goals, as the program development continues we will also find new goals that we want to achieve.  These will be included below as updates to this post.

Planning a New Program

BEAM is receiving funding to develop a new program: a non-residential program in New York City that will reach students from a younger age, beginning the summer after 6th grade.  Students will learn mathematical reasoning, build basic mathematical skills, and become part of an intellectual community.

Now comes of the work of designing that program for a launch this summer.  I’m going to do that design here, on the blog, so that others can follow along with the process of creating a new program and see the ideas get developed and change over time.

To begin, I’ve created an outline of the major topics I plan to think through.  Each of these bullet points will become a link to a post.  Please note that both this post and all of the other posts in the series are likely to evolve over time.  They’re likely to get edited to reflect the final state of thinking as we move to launch.

Big Picture

Before the Summer

  • How is the program communicated to schools and students?
  • How are students selected for the program?
  • How will we hire staff?

Curriculum

Social Environment

  • How do we create a vibrant community?
  • What structures do we need to manage student behavior?

After the Summer

  • What, if any, additional support is provided to students?
  • How does this connect to the existing BEAM program?

Logistics

The program will be known (for now) as BEAM 6, and all posts about it will be labeled as such.

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.